Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{3},|\vec{b}|=5, \vec{b} \cdot \vec{c}=10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$. If $\vec{a}$ is perpendicular to the vector $\vec{b} \times \vec{c}$, then $|\vec{a} \times(\vec{b} \times \vec{c})|$ is equal to ______.

JEE Main · 2020 · 09 Jan 2020 · Shift 2 · Vector Algebra

Let

a, b

and

c

be three vectors such that

∣ a ∣ = 3, ∣ b ∣ = 5, b \cdot c = 10

and the angle between

b

and

c

\frac{π}{3}

. If

a

is perpendicular to the vector

b \times c

, then

∣ a \times (b \times c) ∣

is equal to ______.

Answer: 30

From

b \cdot c = ∣ b ∣∣ c ∣ cos \frac{π}{3}

10 = 5 \cdot ∣ c ∣ \cdot \frac{1}{2}

gives

∣ c ∣ = 4

. Since

a

is perpendicular to

b \times c

, the angle between them is

9 0^{\circ}

, so

∣ a \times (b \times c) ∣ = ∣ a ∣ ∣ b \times c ∣ = ∣ a ∣ ∣ b ∣ ∣ c ∣ sin \frac{π}{3} = 3 \cdot 5 \cdot 4 \cdot \frac{3}{2} = 30

JEE Main · 2020 · 09 Jan 2020 · Shift 2 · Integer answer