Question

If the vectors \vec{A} = a\hat{i} + a\hat{j} + 3\hat{k} A =a i ^ +a j ^ ​ +3 k ^ and \vec{B} = a\hat{i}-2\hat{j}-\hat{k} B =a i ^ −2 j ^ ​ − k ^ are perpendicular to each other them the positive value of 'a' is

Answer 1

Given:

A and B are perpendicular vectors;

$\vec{A} = a\hat{i} + a\hat{j} + 3\hat{k}$

$\vec{B} = a\hat{i}-2\hat{j}-\hat{k}$

To find:

Value of "a".

Diagram:

$\boxed{\setlength{\unitlength}{1cm}\begin{picture}(6,6)\put(3,2){\vector(1,1){1}}\put(3,2){\vector(-1,1){2}}\put(4,3.5){ \vec{A} }\put(1,4.25){ \vec{B} }\put(3,2.5){ 90^{\circ} }\end{picture}}$

Calculation:

For perpendicular vectors , the scalar (dot) product comes as zero.

$\sf{ \therefore \: \vec{A} \: . \: \vec{B} = 0}$

$\sf{ \implies \: (a \hat{i} + a \hat{j} + 3 \hat{k})\: . \: (a \hat{i} - 2 \hat{j} - \hat{k})= 0}$

$\sf{ \implies \: {a}^{2} - 2a - 3 = 0}$

$\sf{ \implies \: {a}^{2} - 3a + a - 3 = 0}$

$\sf{ \implies \: a(a - 3) + 1( a - 3) = 0}$

$\sf{ \implies \: (a + 1)(a - 3) = 0}$

Either a = -1 or a = +3

Since the question asks for only positive value, final answer is:

$\boxed{ \rm{ \: a = + 3}}$