If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$ is ______.

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Vector Algebra

a

and

b

are unit vectors, then the greatest value of

3 ∣ a + b ∣ + ∣ a - b ∣

is ______.

Answer: 4

Let

θ

be the angle between the unit vectors, so

∣ a + b ∣ = 2 + 2 cos θ = 2 cos \frac{θ}{2}

and

∣ a - b ∣ = 2 - 2 cos θ = 2 sin \frac{θ}{2}

. The expression becomes

3 (2 cos \frac{θ}{2}) + 2 sin \frac{θ}{2} = 2 (3 cos \frac{θ}{2} + sin \frac{θ}{2})

, and a term of the form

p cos ϕ + q sin ϕ

has maximum

p^{2} + q^{2}

. Here that maximum is

3 + 1 = 2

, so the greatest value is

2 \times 2 = 4

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Integer answer