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#### Question 1:

Evaluate the following:
(i)

(ii)

#### Question 2:

Find , when
(i)
(ii)
(iii)

(iii)

= 2(−4 − 1) −3(2 + 3) + (1 + (−6))

= 2(−5) − 3(5) + 1(−5)

= −10 − 15 − 5

i.e

#### Question 3:

Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
(i)

(ii)

(iii)

(iv)

#### Question 4:

Show the each of the following triads of vectors are coplanar:
(i)

(ii)

(iii)

#### Question 5:

Find the value of λ so that the following vectors are coplanar:
(i)

(ii)

(iii)

(iv)

#### Question 6:

Show that the four points having position vectors are not coplanar.

#### Question 7:

Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.

#### Question 8:

Show that four points whose position vectors are are coplanar.

DISCLAIMER: Given points are not coplanar.

#### Question 9:

Find the value of λ for which the four points with position vectors are coplanar.

Prove that:

#### Question 11:

are the position vectors of points A, B and C respectively, prove that: is a vector perpendicular to the plane of triangle ABC.

#### Question 12:

(i) If c1 = 1 and c2 = 2, find c3 which makes coplanar.

(ii) If c2 = −1 and c3 = 1, show that no value of c1 can make coplanar.

#### Question 13:

Find λ for which the points A (3, 2, 1), B (4, λ, 5), C (4, 2, −2) and D (6, 5, −1) are coplanar.

#### Question 14:

If four points A, B, C and with position vectors 4$\stackrel{^}{i}+$3$\stackrel{^}{j}+$3$\stackrel{^}{k}$, 5$\stackrel{^}{i}+$ $x\stackrel{^}{j}+$ 7$\stackrel{^}{k}$, 5$\stackrel{^}{i}+$3$\stackrel{^}{j}$ and  respectively are coplanar, then find the value of x.

Let  and $\stackrel{\to }{\mathrm{OD}}=7\stackrel{^}{i}+6\stackrel{^}{j}+\stackrel{^}{k}$.

$\therefore \stackrel{\to }{\mathrm{AB}}=\left(5\stackrel{^}{i}+x\stackrel{^}{j}+7\stackrel{^}{k}\right)-\left(4\stackrel{^}{i}+3\stackrel{^}{j}+3\stackrel{^}{k}\right)=\stackrel{^}{i}+\left(x-3\right)\stackrel{^}{j}+4\stackrel{^}{k}$

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

$\stackrel{\to }{\mathrm{AD}}=\left(7\stackrel{^}{i}+6\stackrel{^}{j}+\stackrel{^}{k}\right)-\left(4\stackrel{^}{i}+3\stackrel{^}{j}+3\stackrel{^}{k}\right)=3\stackrel{^}{i}+3\stackrel{^}{j}-2\stackrel{^}{k}$

Since the given four points are coplanar, so the vectors  and $\stackrel{\to }{\mathrm{AD}}$ are also coplanar.

$\therefore \left[\begin{array}{ccc}\stackrel{\to }{\mathrm{AB}}& \stackrel{\to }{\mathrm{AC}}& \stackrel{\to }{\mathrm{AD}}\end{array}\right]=0$

$⇒\left|\begin{array}{ccc}1& x-3& 4\\ 1& 0& -3\\ 3& 3& -2\end{array}\right|=0\phantom{\rule{0ex}{0ex}}⇒1\left(0+9\right)-\left(x-3\right)\left(-2+9\right)+4\left(3-0\right)=0\phantom{\rule{0ex}{0ex}}⇒9-7x+21+12=0\phantom{\rule{0ex}{0ex}}⇒7x=42\phantom{\rule{0ex}{0ex}}⇒x=6$
Thus, the value of x is 6.

#### Question 15:

Find the volume of the parallelopiped whose adjacent edges are represented by , where .

Given: .
Now,
$2\stackrel{\to }{a}=2\left(\stackrel{^}{i}-\stackrel{^}{j}+2\stackrel{^}{k}\right)=2\stackrel{^}{i}-2\stackrel{^}{j}+4\stackrel{^}{k}\phantom{\rule{0ex}{0ex}}-\stackrel{\to }{b}=-3\stackrel{^}{i}-4\stackrel{^}{j}+5\stackrel{^}{k}\phantom{\rule{0ex}{0ex}}3\stackrel{\to }{c}=3\left(2\stackrel{^}{i}-\stackrel{^}{j}+3\stackrel{^}{k}\right)=6\stackrel{^}{i}-3\stackrel{^}{j}+9\stackrel{^}{k}$
Volume of parallelopiped with adjacent sides  is

Thus volume of Parallelopiped =
= 24 cubic units

#### Question 1:

If lies in the plane of vectors , then which of the following is correct?
(a)

(b)

(c)

(d)

(a)

#### Question 2:

The value of
(a) 0
(b) 1
(c) 6
(d) none of these

(a) 0

#### Question 3:

If are three non-coplanar mutually perpendicular unit vectors, then is
(a) ± 1
(b) 0
(c) −2
(d) 2

#### Question 4:

If $\stackrel{\to }{r}·\stackrel{\to }{a}=\stackrel{\to }{r}·\stackrel{\to }{b}=\stackrel{\to }{r}·\stackrel{\to }{c}=0$ for some non-zero vector $\stackrel{\to }{r},$ then the value of is
(a) 2
(b) 3
(c) 0
(d) none of these

(c) 0

#### Question 5:

For any three vectors the expression equals
(a)

(b)

(c)

(d) none of these

(d) none of these

#### Question 6:

If are non-coplanar vectors, then is equal to
(a) 0
(b) 2
(c) 1
(d) none of these

(a) 0

#### Question 7:

Let be three non-zero vectors such that is a unit vector perpendicular to both . If the angle between is $\frac{\mathrm{\pi }}{6},$ then ${\left|\begin{array}{ccc}{a}_{1}& {a}_{2}& {a}_{3}\\ {b}_{1}& {b}_{2}& {b}_{3}\\ {c}_{1}& {c}_{2}& {c}_{3}\end{array}\right|}^{2}$is equal to

(a) 0
(b) 1
(c)

(d)

(c)

#### Question 8:

If then the volume of the parallelopiped with conterminous edges is
(a) 2
(b) 1
(c) −1
(d) 0

Disclaimer: None of the given options is correct.

If then λ + μ =
(a) 6
(b) −6
(c) 10
(d) 8

(a) 6

(a)

(b)

(c)

(d)

#### Question 11:

If the vectors are coplanar, then m =
(a) 0
(b) 38
(c) −10
(d) 10

#### Question 12:

For non-zero vectors the relation holds good, if
(a)

(b)

(c)

(d)

(a) 0

(b)

(c)

(d)

#### Question 14:

If are three non-coplanar vectors, then equals

(a) 0

(b)

(c)

(d)

is equal to

(a)

(b)

(c)

(d) 0

#### Question 16:

The vectors  are coplanar, if λ =
(a) –2
(b) 0
(c) 1
(d) –1

Hence, the correct answer is option A.

#### Question 17:

The vector  are coplanar, if the value of $\lambda$ is
(a) -2
(b) 0
(c) 2
(d) any real number

Given:  are coplanar.
if  are coplanar

$⇒\left|\begin{array}{ccc}3& -1& 2\\ 2& 1& 3\\ 1& \lambda & -1\end{array}\right|=0\phantom{\rule{0ex}{0ex}}⇒3\left(-1-3\lambda \right)+1\left(-2-3\right)+2\left(2\lambda -1\right)=0\phantom{\rule{0ex}{0ex}}⇒-3-9\lambda -5+4\lambda -2=0\phantom{\rule{0ex}{0ex}}⇒-5\lambda -10=0\phantom{\rule{0ex}{0ex}}⇒\lambda =-2$
Hence, the correct answer is option A.

#### Question 2:

If  are three vectors such that  = _______________.

#### Question 3:

If  are non-coplanar vectors, then  = _______________.

#### Question 4:

For any three vectors  the value of  is _____________.

For any three vectors  ;
$\stackrel{\to }{a}·\left\{\left(\stackrel{\to }{b}+\stackrel{\to }{c}\right)×\left(\stackrel{\to }{a}+\stackrel{\to }{b}+\stackrel{\to }{c}\right)\right\}$

#### Question 5:

The value of  is _________________.

#### Question 6:

For any two vectors  = ________________.

For any two vectors
$\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right).\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)+\left(\stackrel{\to }{a}.\stackrel{\to }{b}\right)$

#### Question 7:

​If non-coplanar vectors  from a parallelopiped of volume 6 cubic units, then the values of  are _______________.

#### Question 8:

If three non-coplanar vectors  from a parallelopiped of volume 8 cubic units, then the values of  are _________________.

Volume of parallelopiped is 8 cubic units

#### Question 9:

If  are non-coplanar vectors, then vectors  from a parallelopiped whose volume is ______________.

#### Question 10:

Let  be three non-coplanar vectors such that  Then the height the parallelopiped formed by  as two adjacent edges of the base, is ____________.

Given

$\left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|=6$

when  are two adjacent edges of the base

$\therefore$ Area of Base is $\left|\stackrel{\to }{a}×\stackrel{\to }{b}\right|=6$.

i.e c will represent height

Since Volume is given by

i.e height of the parallelepiped is 4.

#### Question 11:

If  for some non-zero vector  is  ____________.

Given for same non-zero vector $\stackrel{\to }{r},$

#### Question 1:

Write the value of

#### Question 2:

Write the value of

#### Question 3:

Write the value of

#### Question 4:

Find the values of 'a' for which the vectors are coplanar.

#### Question 5:

Find the volume of the parallelopiped with its edges represented by the vectors

#### Question 6:

If are non-collinear vectors, then find the value of

#### Question 7:

If the vectors (sec2 A) are coplanar, then find the value of cosec2 A + cosec2 B + cosec2 C.

#### Question 8:

For any two vectors of magnitudes 3 and 4 respectively, write the value of

#### Question 9:

If then find the value of λ + μ.

#### Question 10:

If are non-coplanar vectors, then find the value of

#### Question 11:

Find $\stackrel{\to }{a}.\left(\stackrel{\to }{b}×\stackrel{\to }{c}\right)$, if and $\stackrel{\to }{c}=3\stackrel{^}{i}+\stackrel{^}{j}+2\stackrel{^}{k}$.                    [CBSE 2014]

The given vectors are and $\stackrel{\to }{c}=3\stackrel{^}{i}+\stackrel{^}{j}+2\stackrel{^}{k}$.
$\stackrel{\to }{b}×\stackrel{\to }{c}=\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ -1& 2& 1\\ 3& 1& 2\end{array}\right|=3\stackrel{^}{i}+5\stackrel{^}{j}-7\stackrel{^}{k}$
$\therefore \stackrel{\to }{a}.\left(\stackrel{\to }{b}×\stackrel{\to }{c}\right)=\left(2\stackrel{^}{i}+\stackrel{^}{j}+3\stackrel{^}{k}\right).\left(3\stackrel{^}{i}+5\stackrel{^}{j}-7\stackrel{^}{k}\right)=2×3+1×5+3×\left(-7\right)=6+5-21=-10$