Vectors in three dimensions — Question 24
Original QCE Vault practice · 6 marks
Q24 · Practice questionComplex familiar6 marks
QUESTION 24 (6 marks)
Let a=(4,−2,4) and b=(1,2,2). a)Determine a unit vector in the direction of a. b)Determine the scalar projection of a on b. c)Determine the vector projection of a on b. d)Determine the altitude angle of a in exact inverse-trigonometric form. WORKED SOLUTION
Practice marking scheme
6 marksANSWER(a) (2/3,−1/3,2/3); (b) 8/3; (c) (8/9,16/9,16/9); (d) sin−1(2/3). Worked solution
(a) ∣a∣=16+4+16=6, so a=(2/3,−1/3,2/3). (b) a⋅b=4−4+8=8 and ∣b∣=3. The scalar projection is 8/3. (c) projba=b⋅ba⋅bb=98(1,2,2)=(98,916,916). (d) The altitude angle ϕ above the horizontal plane satisfies sinϕ=4/6=2/3, so ϕ=sin−1(2/3). Determines the unit vector.
Determines the scalar projection.
Determines the vector projection.
Determines the altitude angle.
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