Let a⃗ = î + 4ĵ + 2k̂, b⃗ = 3î - 2ĵ + 7k̂ and c⃗ = 2î - ĵ + 4k̂. Find a vector d⃗ which is perpendicular to both a⃗ and b⃗, and c⃗ ⋅ d⃗ = 15.
The required vector is d⃗ = 160/3 î - 5/3 ĵ - 70/3 k̂.
Explanation
The solution follows the method shown in Example 23 of the provided context, where a vector perpendicular to two vectors is found using the cross product. Since the magnitude is not fixed by the perpendicularity condition alone, a scalar parameter λ is introduced. The specific value of the vector is then determined using the dot product condition provided in the question.
Solution Steps
Step 1: Find a vector perpendicular to both a⃗ and b⃗ using the cross product. A vector perpendicular to both a⃗ and b⃗ is given by a⃗ × b⃗. a⃗ × b⃗ = | î ĵ k̂ | | 1 4 2 | | 3 -2 7 | = î(28 - (-4)) - ĵ(7 - 6) + k̂(-2 - 12) = 32î - ĵ - 14k̂
Step 2: Express the vector d⃗ in terms of a scalar λ. Since d⃗ is perpendicular to both a⃗ and b⃗, it is parallel to a⃗ × b⃗. Let d⃗ = λ(32î - ĵ - 14k̂).
Step 3: Use the condition c⃗ ⋅ d⃗ = 15 to find λ. Given c⃗ = 2î - ĵ + 4k̂. c⃗ ⋅ d⃗ = (2î - ĵ + 4k̂) ⋅ λ(32î - ĵ - 14k̂) = 15 λ [2(32) + (-1)(-1) + 4(-14)] = 15 λ [64 + 1 - 56] = 15 9λ = 15 λ = 15/9 = 5/3
Step 4: Calculate the final vector d⃗. Substituting λ = 5/3 into the expression for d⃗: d⃗ = (5/3)(32î - ĵ - 14k̂) d⃗ = 160/3 î - 5/3 ĵ - 70/3 k̂