Question

For two vectors `veca` and `vecb`

Assertion (A): `|vecaxxvecb|^2+(veca*vecb)^2=|veca|^2|vecb|^2`

Reason (R): `|vecaxxvecb|=(veca*vecb)tantheta,(theta≠pi/2)`

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

• Assertion (A) is true, but Reason (R) is false.

• Assertion (A) is false, but Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation:

`|vecaxxvecb|=|veca||vecb|sin theta`

`|vecaxxvecb|^2=|veca|^2|vecb|^2sin^2theta`

`|veca*vecb|=|veca||vecb|costheta`

`|vecaxxvecb|^2=|veca|^2|vecb|^2cos^2theta`

`|vecaxxvecb|^2+(veca*vecb)^2=|veca|^2|vecb|^2sin^2theta+|veca|^2|vecb|^2cos^2theta`

= `|veca|^2|vecb|^2sin^2+cos^2theta`

Since sin²θ + cos²θ = 1, we get:

`|vecaxxvecb|^2+(veca*vecb)^2=|veca|^2|vecb|^2`

`|vecaxxvecb|/(veca*vecb)=(|veca||vecb|sintheta)/(|veca||vecb|costheta)`

= tanθ

`|vecaxxvecb|=(veca*vecb)tanθ`