Rd Sharma XII Vol 2 2020 2021 Solutions for Class 12 Commerce Maths Chapter 6 Vector Or Cross Product are provided here with simple step-by-step explanations. These solutions for Vector Or Cross Product are extremely popular among Class 12 Commerce students for Maths Vector Or Cross Product Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma XII Vol 2 2020 2021 Book of Class 12 Commerce Maths Chapter 6 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma XII Vol 2 2020 2021 Solutions. All Rd Sharma XII Vol 2 2020 2021 Solutions for class Class 12 Commerce Maths are prepared by experts and are 100% accurate.

Question 2:

(i) If find the value of

(ii) If find the magnitude of

Question 3:

(i) Find a unit vector perpendicular to both the vectors

(ii) Find a unit vector perpendicular to the plane containing the vectors

Question 4:

Find the magnitude of

Question 7:

(i) Find a vector of magnitude 49, which is perpendicular to both the vectors

(ii) Find a vector whose length is 3 and which is perpendicular to the vector

Question 8:

Find the area of the parallelogram determined by the vectors:
(i)

(ii)

(iii)

(iv)

Question 9:

Find the area of the parallelogram whose diagonals are:
(i)

(ii)

(iii)

(iv)

Disclaimer: The answer given for (iii) and (iv) in the textbook is incorrect.

Question 10:

If compute and verify that these are not equal.

Question 12:

Given being a right handed orthogonal system of unit vectors in space, show that is also another system.

Question 14:

Find the angle between two vectors , if

Question 15:

If then show that where m is any scalar.

Question 16:

If find the angle between

Question 17:

What inference can you draw if

Question 18:

If are three unit vectors such that Show that form an orthonormal right handed triad of unit vectors.

Question 19:

Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B and C are A (3, −1, 2), B (1, −1, −3) and C (4, −3, 1).

Question 20:

If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that $\stackrel{\to }{BC}+\stackrel{\to }{CA}+\stackrel{\to }{AB}=\stackrel{\to }{0}$ and deduce that

Question 21:

If then find Verify that are perpendicular to each other.

Question 22:

If are unit vectors forming an angle of 30°; find the area of the parallelogram having as its diagonals.

Question 23:

For any two vectors , prove that .

Question 24:

Define and prove that tan θ, where θ is the angle between .

Question 26:

Find the area of the triangle formed by O, A, B when

Question 27:

(i) Let Find a vector which is perpendicular to both

(ii) Let . Find a vector   which is perpendicular to both .

(i)

Disclaimer: The question should contain

(ii)

Question 28:

Find a unit vector perpendicular to each of the vectors

Question 29:

Using vectors find the area of the triangle with vertices, A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).

Question 30:

If are three vectors, find the area of the parallelogram having diagonals $\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)$ and $\left(\stackrel{\to }{b}+\stackrel{\to }{c}\right)$.     [CBSE 2014]

It is given that .

$\stackrel{\to }{a}+\stackrel{\to }{b}=\left(2\stackrel{^}{i}-3\stackrel{^}{j}+\stackrel{^}{k}\right)+\left(-\stackrel{^}{i}+\stackrel{^}{k}\right)=\stackrel{^}{i}-3\stackrel{^}{j}+2\stackrel{^}{k}$

We know that the area of parallelogram is $\frac{1}{2}\left|\stackrel{\to }{{d}_{1}}×\stackrel{\to }{{d}_{2}}\right|$, where $\stackrel{\to }{{d}_{1}}$ and $\stackrel{\to }{{d}_{2}}$ are the diagonal vectors.

Now,

$\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)×\left(\stackrel{\to }{b}+\stackrel{\to }{c}\right)=\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ 1& -3& 2\\ -1& 2& 0\end{array}\right|=-4\stackrel{^}{i}-2\stackrel{^}{j}-\stackrel{^}{k}$

∴ Area of the parallelogram having diagonals $\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)$ and $\left(\stackrel{\to }{b}+\stackrel{\to }{c}\right)$

Thus, the required area of the parallelogram is $\frac{\sqrt{21}}{2}$ square units.

Question 31:

The two adjacent sides of a parallelogram are Find the unit vector parallel to one of its diagonals. Also, find its area.

Question 32:

If either Is the converse true? Justify your answer with an example.

Question 33:

If then verify that

Question 34:

Using vectors, find the area of the triangle with vertices:
(i) A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)
(ii) A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1)                                   [CBSE 2011, NCERT EXEMPLAR]

(i) The vertices of the triangle are A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

Position vector of A = $\stackrel{^}{i}+\stackrel{^}{j}+2\stackrel{^}{k}$

Position vector of B = $2\stackrel{^}{i}+3\stackrel{^}{j}+5\stackrel{^}{k}$

Position vector of C = $\stackrel{^}{i}+5\stackrel{^}{j}+5\stackrel{^}{k}$

$\stackrel{\to }{\mathrm{AB}}=\left(2\stackrel{^}{i}+3\stackrel{^}{j}+5\stackrel{^}{k}\right)-\left(\stackrel{^}{i}+\stackrel{^}{j}+2\stackrel{^}{k}\right)=\stackrel{^}{i}+2\stackrel{^}{j}+3\stackrel{^}{k}$

$\stackrel{\to }{\mathrm{AC}}=\left(\stackrel{^}{i}+5\stackrel{^}{j}+5\stackrel{^}{k}\right)-\left(\stackrel{^}{i}+\stackrel{^}{j}+2\stackrel{^}{k}\right)=4\stackrel{^}{j}+3\stackrel{^}{k}$

Now,

$\stackrel{\to }{\mathrm{AB}}×\stackrel{\to }{\mathrm{AC}}=\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ 1& 2& 3\\ 0& 4& 3\end{array}\right|=-6\stackrel{^}{i}-3\stackrel{^}{j}+4\stackrel{^}{k}$

∴ Area of ∆ABC = $\frac{1}{2}\left|\stackrel{\to }{\mathrm{AB}}×\stackrel{\to }{\mathrm{AC}}\right|$

(ii) The vertices of the triangle are A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).

Position vector of A = $\stackrel{^}{i}+2\stackrel{^}{j}+3\stackrel{^}{k}$

Position vector of B = $2\stackrel{^}{i}-\stackrel{^}{j}+4\stackrel{^}{k}$

Position vector of C = $4\stackrel{^}{i}+5\stackrel{^}{j}-\stackrel{^}{k}$

$\stackrel{\to }{\mathrm{AB}}=\left(2\stackrel{^}{i}-\stackrel{^}{j}+4\stackrel{^}{k}\right)-\left(\stackrel{^}{i}+2\stackrel{^}{j}+3\stackrel{^}{k}\right)=\stackrel{^}{i}-3\stackrel{^}{j}+\stackrel{^}{k}$

$\stackrel{\to }{\mathrm{AC}}=\left(4\stackrel{^}{i}+5\stackrel{^}{j}-\stackrel{^}{k}\right)-\left(\stackrel{^}{i}+2\stackrel{^}{j}+3\stackrel{^}{k}\right)=3\stackrel{^}{i}+3\stackrel{^}{j}-4\stackrel{^}{k}$

Now,

$\stackrel{\to }{\mathrm{AB}}×\stackrel{\to }{\mathrm{AC}}=\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ 1& -3& 1\\ 3& 3& -4\end{array}\right|=9\stackrel{^}{i}+7\stackrel{^}{j}+12\stackrel{^}{k}$

∴ Area of ∆ABC = $\frac{1}{2}\left|\stackrel{\to }{\mathrm{AB}}×\stackrel{\to }{\mathrm{AC}}\right|$

Question 35:

Find all vectors of magnitude $10\sqrt{3}$ that are perpendicular to the plane of $\stackrel{^}{i}+2\stackrel{^}{j}+\stackrel{^}{k}$ and $-\stackrel{^}{i}+3\stackrel{^}{j}+4\stackrel{^}{k}$.                  [NCERT EXEMPLAR]

Let $\stackrel{\to }{a}=\stackrel{^}{i}+2\stackrel{^}{j}+\stackrel{^}{k}$ and $\stackrel{\to }{b}=-\stackrel{^}{i}+3\stackrel{^}{j}+4\stackrel{^}{k}$.

Unit vectors perpendicular to both $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ = $±\frac{\stackrel{\to }{a}×\stackrel{\to }{b}}{\left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|}$

Now,
$\stackrel{\to }{a}×\stackrel{\to }{b}=\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ 1& 2& 1\\ -1& 3& 4\end{array}\right|=5\stackrel{^}{i}-5\stackrel{^}{j}+5\stackrel{^}{k}\phantom{\rule{0ex}{0ex}}\therefore \left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|=\left|5\stackrel{^}{i}-5\stackrel{^}{j}+5\stackrel{^}{k}\right|=\sqrt{{5}^{2}+{\left(-5\right)}^{2}+{5}^{2}}=\sqrt{75}=5\sqrt{3}$

Unit vectors perpendicular to both $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ = $±\frac{5\stackrel{^}{i}-5\stackrel{^}{j}+5\stackrel{^}{k}}{5\sqrt{3}}=±\frac{\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}}{\sqrt{3}}$

∴ Required vectors = $10\sqrt{3}\left(±\frac{\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}}{\sqrt{3}}\right)=±10\left(\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}\right)$

Thus, the vectors of magnitude $10\sqrt{3}$ that are perpendicular to the plane of $\stackrel{^}{i}+2\stackrel{^}{j}+\stackrel{^}{k}$ and $-\stackrel{^}{i}+3\stackrel{^}{j}+4\stackrel{^}{k}$ are $±10\left(\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}\right)$.

Question 36:

The two adjacent sides of a parallelogram are . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

The two adjacent sides of a parallelogram are .
Suppose

Then any one diagonal of a parallelogram is  $\stackrel{\to }{P}=\stackrel{\to }{a}+\stackrel{\to }{b}$.

Therefore, unit vector along the diagonal is $\frac{\stackrel{\to }{P}}{\left|\stackrel{\to }{P}\right|}=\frac{4\stackrel{^}{i}-2\stackrel{^}{j}-2\stackrel{^}{k}}{\sqrt{16+4+4}}=\frac{2\stackrel{^}{i}-\stackrel{^}{j}-\stackrel{^}{k}}{\sqrt{6}}$.
Another diagonal of a parallelogram is $\stackrel{\to }{P\text{'}}=\stackrel{\to }{b}-\stackrel{\to }{a}$.
$\stackrel{\to }{P}\text{'}=\stackrel{\to }{b}-\stackrel{\to }{a}\phantom{\rule{0ex}{0ex}}=2\stackrel{^}{i}+2\stackrel{^}{j}+3\stackrel{^}{k}-2\stackrel{^}{i}+4\stackrel{^}{j}+5\stackrel{^}{k}\phantom{\rule{0ex}{0ex}}=6\stackrel{^}{j}+8\stackrel{^}{k}$
Therefore, unit vector along the diagonal is $\frac{\stackrel{\to }{P\text{'}}}{\left|\stackrel{\to }{P\text{'}}\right|}=\frac{6\stackrel{^}{j}+8\stackrel{^}{k}}{\sqrt{36+64}}=\frac{6\stackrel{^}{j}+8\stackrel{^}{k}}{10}=\frac{3\stackrel{^}{j}+4\stackrel{^}{k}}{5}$.
Now,
$\stackrel{\to }{P}×\stackrel{\to }{P\text{'}}=\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ 4& -2& -2\\ 0& 6& 8\end{array}\right|\phantom{\rule{0ex}{0ex}}=\stackrel{^}{i}\left(-16+12\right)-\stackrel{^}{j}\left(32-0\right)+\stackrel{^}{k}\left(24-0\right)\phantom{\rule{0ex}{0ex}}=-4\stackrel{^}{i}-32\stackrel{^}{j}+24\stackrel{^}{k}$
Area of parallelogram = $\frac{\left|\stackrel{\to }{P}×\stackrel{\to }{P\text{'}}\right|}{2}=\frac{\sqrt{16+1024+576}}{2}=\frac{\sqrt{1616}}{2}=\frac{4\sqrt{101}}{2}=2\sqrt{101}$ square units

Question 37:

If ${\left|\stackrel{\to }{\mathrm{a}}×\stackrel{\to }{\mathrm{b}}\right|}^{2}+{\left|\stackrel{\to }{\mathrm{a}}·\stackrel{\to }{\mathrm{b}}\right|}^{2}=400$ and $\left|\stackrel{\to }{\mathrm{a}}\right|=5,$ then write the value of $\left|\stackrel{\to }{\mathrm{b}}\right|.$

${\left|\stackrel{\mathit{\to }}{\mathit{a}}×\stackrel{\to }{b}\right|}^{2}+{\left|\stackrel{\mathit{\to }}{\mathit{a}}·\stackrel{\to }{b}\right|}^{2}=400\phantom{\rule{0ex}{0ex}}⇒{\left\{\left|\stackrel{\to }{a}\right|\left|\stackrel{\to }{b}\right|\mathrm{sin}\theta \right\}}^{2}+{\left\{\left|\stackrel{\to }{a}\right|\left|\stackrel{\to }{b}\right|\mathrm{cos}\theta \right\}}^{2}=400\phantom{\rule{0ex}{0ex}}⇒{\left|\stackrel{\to }{a}\right|}^{2}{\left|\stackrel{\to }{b}\right|}^{2}{\mathrm{sin}}^{2}\theta +{\left|\stackrel{\to }{a}\right|}^{2}{\left|\stackrel{\to }{b}\right|}^{2}{\mathrm{cos}}^{2}\theta =400\phantom{\rule{0ex}{0ex}}⇒{\left|\stackrel{\to }{a}\right|}^{2}{\left|\stackrel{\to }{b}\right|}^{2}=400\phantom{\rule{0ex}{0ex}}⇒25×{\left|\stackrel{\to }{b}\right|}^{2}=400\phantom{\rule{0ex}{0ex}}⇒{\left|\stackrel{\to }{b}\right|}^{2}=16\phantom{\rule{0ex}{0ex}}⇒\left|\stackrel{\to }{b}\right|=4\phantom{\rule{0ex}{0ex}}$

Question 38:

If θ is the angle between two vectors .

Let $\stackrel{\to }{a}=\stackrel{^}{i}-2\stackrel{^}{j}+3\stackrel{^}{k}$ and $\stackrel{\to }{b}=3\stackrel{^}{i}-2\stackrel{^}{j}+\stackrel{^}{k}$. If θ is the angle between them. Then,
$\mathrm{cos}\theta =\frac{\stackrel{\to }{a}·\stackrel{\to }{b}}{\left|\stackrel{\to }{a}\right|\left|\stackrel{\to }{b}\right|}$
Now,

$\therefore \mathrm{cos}\theta =\frac{\stackrel{\to }{a}·\stackrel{\to }{b}}{\left|\stackrel{\to }{a}\right|\left|\stackrel{\to }{b}\right|}\phantom{\rule{0ex}{0ex}}⇒\mathrm{cos}\theta =\frac{10}{\sqrt{14}\sqrt{14}}=\frac{10}{14}=\frac{5}{7}$
Now,

Question 1:

If is any vector, then
(a) ${\stackrel{\to }{a}}^{2}$

(b) $2{\stackrel{\to }{a}}^{2}$

(c) $3{\stackrel{\to }{a}}^{2}$

(d) $4{\stackrel{\to }{a}}^{2}$

(b) $2{\stackrel{\to }{a}}^{2}$

Question 2:

If and then
(a)

(b)

(c)

(d) none of these

(a)

Question 3:

The vector is to be written as the sum of a vector parallel to and a vector perpendicular to . Then
(a) $\frac{3}{2}\left(\stackrel{^}{i}+\stackrel{^}{j}\right)$

(b) $\frac{2}{3}\left(\stackrel{^}{i}+\stackrel{^}{j}\right)$

(c) $\frac{1}{2}\left(\stackrel{^}{i}+\stackrel{^}{j}\right)$

(d) $\frac{1}{3}\left(\stackrel{^}{i}+\stackrel{^}{j}\right)$

(a) $\frac{3}{2}\left(\stackrel{^}{i}+\stackrel{^}{j}\right)$

Question 4:

The unit vector perpendicular to the plane passing through points is
(a) $2\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}$

(b) $\sqrt{6}\left(2\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}\right)$

(c) $\frac{1}{\sqrt{6}}\left(2\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}\right)$

(d) $\frac{1}{6}\left(2\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}\right)$

(c) $\frac{1}{\sqrt{6}}\left(2\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}\right)$

Question 5:

If represent the diagonals of a rhombus, then
(a)

(b)

(c)

(d)

(b)

Question 6:

Vectors are inclined at angle θ = 120°. If then is equal to
(a) 300
(b) 325
(c) 275
(d) 225

(a) 300

Question 7:

If then a unit vector normal to the vectors is
(a) $\stackrel{^}{i}$

(b) $\stackrel{^}{j}$

(c) $\stackrel{^}{k}$

(d) none of these

(a) $\stackrel{^}{i}$

Question 8:

A unit vector perpendicular to both is

(a) $\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}$

(b) $\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}$

(c) $\frac{1}{\sqrt{3}}\left(\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}\right)$

(d) $\frac{1}{\sqrt{3}}\left(\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}\right)$

(d) $\frac{1}{\sqrt{3}}\left(\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}\right)$

Disclaimer: The answer given for this question in the textbook is incorrect.

Question 9:

If is
(a) $10\stackrel{^}{i}+2\stackrel{^}{j}+11\stackrel{^}{k}$

(b) $10\stackrel{^}{i}+3\stackrel{^}{j}+11\stackrel{^}{k}$

(c) $10\stackrel{^}{i}-3\stackrel{^}{j}+11\stackrel{^}{k}$

(d) $10\stackrel{^}{i}-2\stackrel{^}{j}-10\stackrel{^}{k}$

(b) $10\stackrel{^}{i}+3\stackrel{^}{j}+11\stackrel{^}{k}$

Question 10:

If are unit vectors, then
(a) $\stackrel{^}{i}·\stackrel{^}{j}=1$

(b) $\stackrel{^}{i}·\stackrel{^}{i}=1$

(c) $\stackrel{^}{i}×\stackrel{^}{j}=1$

(d) $\stackrel{^}{i}×\left(\stackrel{^}{j}×\stackrel{^}{k}\right)=1$

(b)  $\stackrel{^}{i}·\stackrel{^}{i}=1$

Question 11:

If θ is the angle between the vectors then sin θ =
(a) $\frac{2}{3}$

(b) $\frac{2}{\sqrt{7}}$

(c) $\frac{\sqrt{2}}{7}$

(d) $\sqrt{\frac{2}{7}}$

(b) $\frac{2}{\sqrt{7}}$

If

(a) 6
(b) 2
(c) 20
(d) 8

(c) 20

The value of is

(a)

(b)

(c)

(d)

(b)

Question 14:

The value of $\stackrel{^}{i}·\left(\stackrel{^}{j}×\stackrel{^}{k}\right)+\stackrel{^}{j}·\left(\stackrel{^}{i}×\stackrel{^}{k}\right)+\stackrel{^}{k}·\left(\stackrel{^}{i}×\stackrel{^}{j}\right),$ is
(a) 0
(b) −1
(c) 1
(d) 3

(c) 1

Question 15:

If θ is the angle between any two vectors , then when θ is equal to

(a) 0
(b) π/4
(c) π/2
(d) π

(b)  π/4

Question 16:

If  then the value of $\left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|$ is
(a) 5
(b) 10
(c) 14
(d) 16

Hence, the correct answer is option D.

Question 17:

The number of vectors of unit length perpendicular to the vectors
(a) one
(b) two
(c) three
(d) infinite

i.e. number of vectors of unit length perpendicular to the vectors  is two.

Hence, the correct answer is option B.

Question 18:

then the value of $\stackrel{\to }{a}·\stackrel{\to }{b}$ is

Hence, the correct answer is option C.

Question 19:

The vectors from origin O to the points A and B are  respectively, then area of triangle OAB is
(a) 340

Hence, the correct answer is option D.

Question 1:

The value of the expression ${\left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|}^{2}+{\left(\stackrel{\to }{a}·\stackrel{\to }{b}\right)}^{2}$ is ______________.

By lagrange's identity,

Question 2:

is equal to ____________.

Question 3:

If  are unit vectors such that $\stackrel{\to }{a}×\stackrel{\to }{b}$ is also a unit vector, then the angle between  is ___________.

Question 4:

For any two vectors  = _________________.

Question 5:

The number of vectors of unit length perpendicular to vectors

The unit vector perpendicular to given by $±\frac{\stackrel{\to }{a}×\stackrel{\to }{b}}{\left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|}$

Question 7:

For any non-zero vector

Question 8:

If  are the position vectors of the vertices A, B and C respectively of a  then area of $∆ABC$ is ____________.

Question 9:

If  are two vectors such that  then the angle between  is _____________.

Question 10:

For any two-collinear vectors , the value of $\stackrel{\to }{a}.\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)$ is ____________.

Question 11:

If  are two non-zero non-collinear vectors such that , then the angle between  is _____________.

Question 12:

If three points with position vectors  are collinear, then

If  are collinear,

Let points A, B, C be collinear, where position vectors are  respectively, since  are parallel vectors,

Question 1:

Define vector product of two vectors.

Question 2:

Write the value $\left(\stackrel{^}{i}×\stackrel{^}{j}\right)·\stackrel{^}{k}+\stackrel{^}{i}·\stackrel{^}{j}.$

Question 3:

Write the value of $\stackrel{^}{i}.\left(\stackrel{^}{j}×\stackrel{^}{k}\right)+\stackrel{^}{j}.\left(\stackrel{^}{k}×\stackrel{^}{i}\right)+\stackrel{^}{k}.\left(\stackrel{^}{j}×\stackrel{^}{i}\right).$

Question 4:

Write the value of $\stackrel{^}{i}.\left(\stackrel{^}{j}×\stackrel{^}{k}\right)+\stackrel{^}{j}.\left(\stackrel{^}{k}×\stackrel{^}{i}\right)+\stackrel{^}{k}.\left(\stackrel{^}{i}×\stackrel{^}{j}\right).$

Question 5:

Write the value of $\stackrel{^}{i}×\left(\stackrel{^}{j}+\stackrel{^}{k}\right)+\stackrel{^}{j}×\left(\stackrel{^}{k}+\stackrel{^}{i}\right)+\stackrel{^}{k}×\left(\stackrel{^}{i}+\stackrel{^}{j}\right).$

Question 6:

Write the expression for the area of the parallelogram having as its diagonals.

Question 7:

For any two vectors write the value of in terms of their magnitudes.

Question 8:

If are two vectors of magnitudes 3 and $\frac{\sqrt{2}}{3}$ respectively such that is a unit vector. Write the angle between

Question 10:

For any two vectors and , find

Question 11:

If are two vectors such that find the angle between.

Question 12:

For any three vectors write the value of

$\stackrel{\to }{a}×\left(\stackrel{\to }{b}+\stackrel{\to }{c}\right)+\stackrel{\to }{b}×\left(\stackrel{\to }{c}+\stackrel{\to }{a}\right)+\stackrel{\to }{c}×\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)\phantom{\rule{0ex}{0ex}}=\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)+\left(\stackrel{\to }{a}×\stackrel{\to }{c}\right)+\left(\stackrel{\to }{b}×\stackrel{\to }{c}\right)+\left(\stackrel{\to }{b}×\stackrel{\to }{a}\right)+\left(\stackrel{\to }{c}×\stackrel{\to }{a}\right)+\left(\stackrel{\to }{c}×\stackrel{\to }{b}\right)\phantom{\rule{0ex}{0ex}}=\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)+\left(\stackrel{\to }{a}×\stackrel{\to }{c}\right)+\left(\stackrel{\to }{b}×\stackrel{\to }{c}\right)-\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)-\left(\stackrel{\to }{a}×\stackrel{\to }{c}\right)-\left(\stackrel{\to }{b}×\stackrel{\to }{c}\right)\phantom{\rule{0ex}{0ex}}=\stackrel{\to }{0}$

Question 13:

For any two vectors

Question 14:

Write the value of $\stackrel{^}{i}×\left(\stackrel{^}{j}×\stackrel{^}{k}\right).$

$\stackrel{^}{i}×\left(\stackrel{^}{j}×\stackrel{^}{k}\right)\phantom{\rule{0ex}{0ex}}=\stackrel{^}{i}×\stackrel{^}{i}\phantom{\rule{0ex}{0ex}}=\stackrel{\to }{0}$

If and then find

Question 16:

Write a unit vector perpendicular to

If and find .

Question 18:

If then write the value of

Question 19:

If are unit vectors such that is also a unit vector, find the angle between .

Question 20:

If are two vectors such that write the angle between

Question 21:

If are unit vectors, then write the value of

Question 22:

If is a unit vector such that

Question 23:

If is a unit vector perpendicular to the vectors write another unit vector perpendicular to

Question 24:

Find the angle between two vectors with magnitudes 1 and 2 respectively and when

Question 25:

Vectors are such that is a unit vector. Write the angle between .

Find λ, if

Question 27:

Write the value of the area of the parallelogram determined by the vectors

Question 28:

Write the value of $\left(\stackrel{^}{i}×\stackrel{^}{j}\right)·\stackrel{^}{k}+\left(\stackrel{^}{j}+\stackrel{^}{k}\right)·\stackrel{^}{j}$

Question 29:

Find a vector of magnitude $\sqrt{171}$ which is perpendicular to both of the vectors $\stackrel{\to }{a}=\stackrel{^}{i}+2\stackrel{^}{j}-3\stackrel{^}{k}$ and $\stackrel{\to }{b}=3\stackrel{^}{i}-\stackrel{^}{j}+2\stackrel{^}{k}$.

The given vectors are $\stackrel{\to }{a}=\stackrel{^}{i}+2\stackrel{^}{j}-3\stackrel{^}{k}$ and $\stackrel{\to }{b}=3\stackrel{^}{i}-\stackrel{^}{j}+2\stackrel{^}{k}$.

Unit vectors perpendicular to both $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ = $±\frac{\stackrel{\to }{a}×\stackrel{\to }{b}}{\left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|}$

Now,
$\stackrel{\to }{a}×\stackrel{\to }{b}=\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ 1& 2& -3\\ 3& -1& 2\end{array}\right|=\stackrel{^}{i}-11\stackrel{^}{j}-7\stackrel{^}{k}\phantom{\rule{0ex}{0ex}}\therefore \left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|=\left|\stackrel{^}{i}-11\stackrel{^}{j}-7\stackrel{^}{k}\right|=\sqrt{{1}^{2}+{\left(-11\right)}^{2}+{\left(-7\right)}^{2}}=\sqrt{1+121+49}=\sqrt{171}$

Unit vectors perpendicular to both $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ = $±\frac{\stackrel{^}{i}-11\stackrel{^}{j}-7\stackrel{^}{k}}{\sqrt{171}}$

∴ Required vectors = $\sqrt{171}\left(±\frac{\stackrel{^}{i}-11\stackrel{^}{j}-7\stackrel{^}{k}}{\sqrt{171}}\right)=±\left(\stackrel{^}{i}-11\stackrel{^}{j}-7\stackrel{^}{k}\right)$

Thus, the vectors of magnitude $\sqrt{171}$ which are perpendicular to both the given vectors are $±\left(\stackrel{^}{i}-11\stackrel{^}{j}-7\stackrel{^}{k}\right)$.

Question 30:

Write the number of vectors of unit length perpendicular to both the vectors .

Unit vectors perpendicular to $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ are $±\left(\frac{\stackrel{\to }{a}×\stackrel{\to }{b}}{\left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|}\right)$.
$\stackrel{\to }{a}×\stackrel{\to }{b}=\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ 2& 1& 2\\ 0& 1& 1\end{array}\right|=-\stackrel{^}{i}-2\stackrel{^}{j}+2\stackrel{^}{k}$

∴ Unit vectors perpendicular to $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ are $±\frac{-\stackrel{^}{i}-2\stackrel{^}{j}+2\stackrel{^}{k}}{\sqrt{{\left(-1\right)}^{2}+{\left(-2\right)}^{2}+{\left(2\right)}^{2}}}=±\left(-\frac{1}{3}\stackrel{^}{i}-\frac{2}{3}\stackrel{^}{j}+\frac{2}{3}\stackrel{^}{k}\right)$

Thus, there are two unit vectors perpendicular to the given vectors.

Question 31:

Write the angle between the vectors $\stackrel{\to }{a}×\stackrel{\to }{b}$ and $\stackrel{\to }{b}×\stackrel{\to }{a}$.

$\stackrel{\to }{b}×\stackrel{\to }{a}$ = $-\stackrel{\to }{a}×\stackrel{\to }{b}$
So, $\stackrel{\to }{a}×\stackrel{\to }{b}$ and $\stackrel{\to }{b}×\stackrel{\to }{a}$ are vectors of same magnitude but opposite in directions.
Thus, the angle between the vectors $\stackrel{\to }{a}×\stackrel{\to }{b}$ and $\stackrel{\to }{b}×\stackrel{\to }{a}$ is 180º.