If a⃗ = 2î + 2ĵ + 3k̂, b⃗ = -î + 2ĵ + k̂ and c⃗ = 3î + ĵ are such that a⃗ + λb⃗ is perpendicular to c⃗, then find the value of λ.
λ = 8
Explanation
This question tests the concept of perpendicular vectors using dot product. When two vectors are perpendicular, their dot product equals zero. The student needs to first compute the vector sum a⃗ + λb⃗ using vector addition and scalar multiplication rules from the context, then apply the perpendicularity condition.
Solution Steps
Step 1: Calculate a⃗ + λb⃗ using vector addition and scalar multiplication. a⃗ + λb⃗ = (2î + 2ĵ + 3k̂) + λ(-î + 2ĵ + k̂) = (2 - λ)î + (2 + 2λ)ĵ + (3 + λ)k̂
Step 2: Apply the condition for perpendicular vectors. Since a⃗ + λb⃗ is perpendicular to c⃗ = 3î + ĵ, their dot product must be zero. (a⃗ + λb⃗) · c⃗ = 0
Step 3: Compute the dot product. (2 - λ)(3) + (2 + 2λ)(1) + (3 + λ)(0) = 0 6 - 3λ + 2 + 2λ = 0 8 - λ = 0
Step 4: Solve for λ. λ = 8