If $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are perpendicular, then $\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})))$ is equal to:

(C) $|\overrightarrow{\mathrm{a}}|^{4} \overrightarrow{\mathrm{~b}}$

(D) $\frac{1}{2}|\vec{a}|^{4} \vec{b}$

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Vector Algebra

a

and

b

are perpendicular, then

a \times (a \times (a \times (a \times b)))

is equal to:

Answer: (C)

Since

a \cdot b = 0

, the triple product expansion gives

a \times (a \times b) = (a \cdot b) a - (a \cdot a) b = - ∣ a ∣^{2} b

Crossing with

a

once more,

a \times (a \times (a \times b)) = a \times (- ∣ a ∣^{2} b) = - ∣ a ∣^{2} (a \times b)

and crossing with

a

a fourth time,

a \times (- ∣ a ∣^{2} (a \times b)) = - ∣ a ∣^{2} (- ∣ a ∣^{2} b) = ∣ a ∣^{4} b

The required product is

∣ a ∣^{4} b

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Single correct option