Test Index

WBJEE 2025 Math Paper

© examsnet.com

Question : 26 of 75

Marks: +1, -0

If '

\theta

' is the angle between two vectors

\overset{\rightarrow}{a}

\overset{\rightarrow}{b}

\left|\overset{\rightarrow}{a}\right|=7,\left|\overset{\rightarrow}{b}\right|=1

\left|\overset{\rightarrow}{a}\times \overset{\rightarrow}{b}\right|^2=k^2-\left(\overset{\rightarrow}{a}\cdot \overset{\rightarrow}{b}\right)^2

, then the values of

k

\theta

are

$k=1, \theta=45^{\circ}$
$k=7, \theta=60^{\circ}$
$k=49, \theta=90^{\circ}$
$k=7$ and $\theta$ is arbitrary

Solution:

We know the identity

|a \times b|^2+(a \cdot b)^2=|a|^2|b|^2 .

Since

|a|=7

and

|b|=1

, the right-hand side is

7^2 \cdot 1^2=49

The problem states

|a \times b|^2=k^2-(a \cdot b)^2 .

Comparing with the identity, we get

k^2=49\;\; \Rightarrow \;\; k=7 .

No further restriction on the angle

\theta

is needed-this relation holds for any

\theta

.

Answer:

k=7

, and

\theta

can be arbitrary. (Option D )

© examsnet.com

Go to Question:

Prev Question

Next Question