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Vector Algebra

Multiple Choice Questions

31.

For any vector $\vec{a}$ , ${|\vec{a} \times \hat{i}|}^{2} + {|\vec{a} \times \hat{j}|}^{2} + {|\vec{a} \times \hat{k}|}^{2}$ = ?

${|\vec{a}|}^{2}$
$2 {|\vec{a}|}^{2}$
$3 {|\vec{a}|}^{2}$
$4 {|\vec{a}|}^{2}$

32.

If the vectors $a \hat{i} + \hat{j} + \hat{k}, \hat{i} + b \hat{j} + \hat{k} and$ $\hat{i} + \hat{j} + c \hat{k}$ $(a, b, c \neq 1) are coplanar, then the value of \frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = ?$

0
1
a + b + c
abc

33.

$Let, \hat{a}, \hat{b} be two unit vectors and θ be the angle between them$ .

What is $\cos (\frac{θ}{2})$ = ?

$\frac{|\hat{a} - \hat{b}|}{2}$
$\frac{|\hat{a} + \hat{b}|}{2}$
$\frac{|\hat{a} - \hat{b}|}{4}$
$\frac{|\hat{a} + \hat{b}|}{4}$

34.

$Let, \hat{a}, \hat{b} be two unit vectors and θ be the angle between them$ .

What is $\sin (\frac{θ}{2})$ = ?

$\frac{|\hat{a} - \hat{b}|}{2}$
$\frac{|\hat{a} + \hat{b}|}{2}$
$\frac{|\hat{a} - \hat{b}|}{4}$
$\frac{|\hat{a} + \hat{b}|}{4}$

35.

What is a vector of unit length orthogonal to both the vectors $\hat{i} + \hat{j} + \hat{k} and 2 \hat{i} + 3 \hat{j} - 3 \hat{k} ?$

$\frac{- 4 \hat{i} + 3 \hat{j} - \hat{k}}{\sqrt{26}}$
$\frac{- 4 \hat{i} + 3 \hat{j} + \hat{k}}{\sqrt{26}}$
$\frac{- 3 \hat{i} + 2 \hat{j} - \hat{k}}{\sqrt{14}}$
$\frac{- 3 \hat{i} + 2 \hat{j} + \hat{k}}{\sqrt{14}}$

36.

If $\vec{a}, \vec{b} and \vec{c}$ are the position vectors of the vertices of an equilateral triangle whose ortho centre is at the origin, then which one of the following is correct ?

$\vec{a} + \vec{b} + \vec{c} = \vec{0}$
$\vec{a} + \vec{b} + \vec{c} = unit vector$
$\vec{a} + \vec{b} = \vec{c}$
$\vec{a} = \vec{b} + \vec{c}$

37.

$Let \vec{a}, \vec{b} and \vec{c} be three vectors such that \vec{a} + \vec{b} + \vec{c} = 0, and |\vec{a}| = 10, |\vec{b}| = 6 and |\vec{c}| = 14 . What is \vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} = ?$

- 332
- 166
0
166

38.

$Let \vec{a} = \hat{i} + \hat{j}, \vec{b} = 3 \hat{i} + 4 \hat{k} and \vec{b} = \vec{c} + \vec{d}, where \vec{c} is parallel to \vec{a} and \vec{d} is perpendicular to \vec{a} . W h a t i s \vec{c} = ?$

$\frac{3 (\hat{i} + \hat{j})}{2}$
$\frac{2 (\hat{i} + \hat{j})}{3}$
$\frac{(\hat{i} + \hat{j})}{2}$
$\frac{(\hat{i} + \hat{j})}{3}$

39.
$Let \vec{a} = \hat{i} + \hat{j}, \vec{b} = 3 \hat{i} + 4 \hat{k} and \vec{b} = \vec{c} + \vec{d}, where \vec{c} is parallel to \vec{a} and \vec{d} is perpendicular to \vec{a} . If \vec{d} = x \hat{i} + y \hat{j} + z \hat{k}, then which of the following equations is / are correct ?$

y - x = 4

2z - 3 = 0

Select the correct answer using the code given below :

1 only

2 only

Both 1 and 2

Neither 1 nor 2

Neither 1 nor 2

$\vec{b} = \vec{c} + \vec{d} \vec{b} = λ \vec{a} + \vec{d} 3 \hat{i} + 4 \vec{k} = λ \hat{i} + λ \hat{j} + x \hat{i} + y \hat{j} + z \hat{k} \vec{a} . \vec{d} = 0 (\hat{i} + \hat{j}) . (x \hat{i} + y \hat{j} + z \hat{k}) = 0 x + y + 0 = 0 x = - y 3 \hat{i} + 4 \hat{k} = (λ + x) \hat{i} + (λ - x) \hat{j} + z \hat{k} (λ + x) = 3 \Rightarrow 2 λ = 3 \Rightarrow λ = \frac{3}{2} λ - x = 0 \Rightarrow x = \frac{3}{2} z = 4 z = 4 \vec{c} = \frac{3}{2} \hat{i} + \frac{3}{2} \hat{j} \vec{d} = \frac{3}{2} \hat{i} - \frac{3}{2} \hat{j} + 4 \hat{k}$

Option (D) is correct.

40.

$Let \vec{a}, \vec{b} and \vec{c} be three vectors such that \vec{a} + \vec{b} + \vec{c}, and [\vec{a}] = 10, [\vec{b}] = 6 and |\vec{c}| = 14 . What is \vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} = ?$

$30 °$
$45 °$
$60 °$
$75 °$