Question

Find the value of \( a \) for which the vectors \( 3 \hat{i}+2 \hat{j}+9 \hat{k} \) and \( \hat{i}+a \hat{j}+3 \hat{k} \) are perpendicular.

Answer 1

Vector 1 = 3i + 2j + 9k, 

Direction Ratios = (a₁, b₁, c₁) = (3, 2, 9)

Vector 2 = 1i + aj + 3k, Direction Ratios = (a₂, b₂, c₂) = (1, a, 3)

(For two vectors to be perpendicular, the condition is,  a₁.a₂+ b₁.b₂ + c₁.c₂ = 0)

∴ (3.1)+(2.a)+(9.3) = 0

⇒ 3 + 2a + 27 = 0

⇒ 2a = -30

⇒ a = -15

Answer 2

3i + 2j + 9k and i + aj + 3k,  if these vectors are perpendicular, the dot product will be Zero

(3)(1) + (2)(a) + (9)(3) = 0

2a = - 30

a = -15