Question

Jake asked Sally to draw a vector perpendicular to the vector 4.0i - 6.0j +2.0k. Which vector would Sally draw?

A) 1.0i-2.0j+1.0k B) 2.0i -3.0j +1.0k C) 2.0i+1.0k D) 1.0i +1.0j+1.0k E) 3.0i -3.0j+3.0k

Answer 1

Two vector will be perpendicular to each other if their dot product is zero.

A.B = 0

given that vector A = 4.0 i - 6.0 j + 2.0 k

Now Suppose vector B = a i + b j + c k

So,

A.B = (4.0 i - 6.0 j + 2.0).(a i + b j + c k)

(Since i.j = j.k = k.i = 0), So

A.B = 4.0*a*(i.i) - 6.0*b*(j.j) + 2.0*c*(k.k)

Since i.i = j.j = k.k = 1, So

A.B = 4a - 6b + 2c

for vector to be perpendicular A.B = 0, So

4a - 6b + 2c = 0

Now all the values of combination (a, b, c) which satisfies above equation will give perpendicular vector of A. So

Option A.

A.B = (4.0 i - 6.0 j + 2.0).(1.0 i - 2.0 j + 1.0 k)

A.B = 4.0*(i.i) + 6.0*2.0*(j.j) + 2.0*1.0*(k.k)

A.B = 4 + 12 + 2 = 18

Option B

A.B = (4.0 i - 6.0 j + 2.0).(2.0 i - 3.0 j + 1.0 k)

A.B = 4.0*2.0*(i.i) + 6.0*3.0*(j.j) + 2.0*1.0*(k.k)

A.B = 8 + 18 + 2 = 28

Option C

A.B = (4.0 i - 6.0 j + 2.0).(2.0 i - 0 j + 1.0 k)

A.B = 4.0*2.0*(i.i) + 6.0*0*(j.j) + 2.0*1.0*(k.k)

A.B = 8 + 0 + 2 = 10

Option D

A.B = (4.0 i - 6.0 j + 2.0).(1.0 i + 1.0 j + 1.0 k)

A.B = 4.0*1.0*(i.i) - 6.0*1.0*(j.j) + 2.0*1.0*(k.k)

A.B = 4  - 6 + 2 = 0

Correct option is D

Option E

A.B = (4.0 i - 6.0 j + 2.0).(3.0 i - 3.0 j + 3.0 k)

A.B = 4.0*3.0*(i.i) + 6.0*3.0*(j.j) + 2.0*3.0*(k.k)

A.B = 12 + 18 + 6 = 36

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