Let the vectors $\vec{a}, \vec{b}, \vec{c}$ be such that $|\vec{a}|=2,|\vec{b}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on $\vec{a}$ is equal to the projection of $\vec{c}$ on $\vec{a}$ and $\vec{b}$ is perpendicular to $\vec{c}$, then the value of $|\vec{a}+\vec{b}-\vec{c}|$ is ____.

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Vector Algebra

Let the vectors

a, b, c

be such that

∣ a ∣ = 2, ∣ b ∣ = 4

and

∣ c ∣ = 4

. If the projection of

b

a

is equal to the projection of

c

a

and

b

is perpendicular to

c

, then the value of

∣ a + b - c ∣

is ____.

Answer: 6

Equal projections on

a

mean

a \cdot b = a \cdot c

, and

b ⊥ c

means

b \cdot c = 0

. Expanding,

∣ a + b - c ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} + ∣ c ∣^{2} + 2 a \cdot b - 2 a \cdot c - 2 b \cdot c

. The last term is zero and the two middle terms cancel because

a \cdot b = a \cdot c

, leaving

4 + 16 + 16 = 36

. Hence

∣ a + b - c ∣ = 6

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Integer answer