Question

If we are given two vectors a and b such that $\left| a \right|$ = 8, $\left| b \right|$ = 3 and $\left| a\times b \right|$ = 12, then the value of a . b is?

(a) 6 or -6
(b) $12\sqrt{3}$ or $-12\sqrt{3}$
(c) 8 or -8
(d) None of these

Answer 1

Hint: If we are given two vector a and b and $\left| a \right|$, $\left| b \right|$ represents the magnitude of these two vectors, then from vector theory, the magnitude of the cross product of these two vector $\left| a\times b \right|$ is given by the formula $\left| a\times b \right|=\left| a \right|\left| b \right|\sin x$ and the dot product of these two vectors a. b is given by the formula $a.b=\left| a \right|\left| b \right|\cos x$. Here, x is the angle between the two vectors a and b. Using these formulas, we can solve this question.

Complete step by step answer:
Before proceeding with the question, we must know all the formulas that will be required to solve this question.

In vector theory, if we are given two vector a and b and $\left| a \right|$, $\left| b \right|$ represents the magnitude of these two vectors, then,

The magnitude of the cross product of the two vectors is given by the formula,

$\left| a\times b \right|=\left| a \right|\left| b \right|\sin x$ . . . . . . . . . . . . (1)

The dot product of the two vectors is given by the formula,

$a.b=\left| a \right|\left| b \right|\cos x$. . . . . . . . . . . . . . (2)

Here x is the angle between the two vectors a and b.

In this question, we are given two vectors a and b such that $\left| a \right|$ = 8, $\left| b \right|$ = 3 and $\left| a\times b \right|$ = 12. Using formula (1), we can say,

$\begin{align}

  & \left| a\times b \right|=\left| a \right|\left| b \right|\sin x \\

 & \Rightarrow 12=3.8.\sin x \\

 & \Rightarrow \sin x=\dfrac{12}{24} \\

 & \Rightarrow \sin x=\dfrac{1}{2} \\

 & \Rightarrow x={{30}^{\circ }},x={{150}^{\circ }} \\

\end{align}$

Since we are required to find the value of a. b, using formula (2), we get,

\[\begin{align}

  & a.b=\left| a \right|\left| b \right|\cos x \\

 & \Rightarrow a.b=8.3.\cos 30,a.b=8.3.\cos 150 \\

 & \Rightarrow a.b=8.3.\dfrac{\sqrt{3}}{2},a.b=8.3.\left( -\dfrac{\sqrt{3}}{2} \right) \\

 & \Rightarrow a.b=12\sqrt{3},a.b=-12\sqrt{3} \\

\end{align}\]

Hence, the answer is option (b).

Note: There is a possibility that one may commit a mistake while writing the value of cos30. It is a common mistake that one write \[\cos 30=\dfrac{1}{2}\] instead of $\cos 30=\dfrac{\sqrt{3}}{2}$ which will lead us to an incorrect answer.