Worksheet: Dot Product in 3D

In this worksheet, we will practice finding the dot product of two vectors in 3D.

Q1:

If A and B are two perpendicular vectors, then AB=.

Q2:

If the dot product of two nonzero vectors is zero, what does this tell us about the two vectors?

  • AThe two vectors are equal in magnitude but opposite in direction.
  • BThey are parallel in opposite directions.
  • CThey are perpendicular.
  • DIt does not indicate anything about the two vectors.
  • EThey are parallel in the same direction.

Q3:

For the unit vectors i, j, k, what is ii?

Q4:

Given that A=6,3,5 and B=7,4,1, determine AB.

Q5:

Given that the coordinates of 𝐴, 𝐵, and 𝐶 are (2,4,2), (2,3,3), and (4,2,5), respectively, determine 𝐴𝐵𝐴𝐶.

Q6:

For what value of 𝑘 are vectors A=7,7𝑘,6 and B=7,3,𝑘 perpendicular?

  • A 4 9 1 5
  • B 4 9 1 5
  • C 7 3
  • D 7 1 5

Q7:

Given that 𝐴𝐵𝐶𝐷 is a square of side 33, determine 𝐴𝐵𝐶𝐴.

Q8:

Find 1,2,3,42,0,1,3.

Q9:

Given that Aijk=57+7 and Bijk=725, determine AB.

Q10:

If A and B are two perpendicular unit vectors, find (3)(2+)ABAB.

Q11:

Which of the following is true of the vectors A=3,7,8 and B=6,1,1?

  • AThey are perpendicular.
  • BThey are neither parallel nor perpendicular.
  • CThey are parallel.

Q12:

If Vijk=32 and Wijk=6+4+2, calculate VW.

Q13:

If A=0,3,1 and Bijk=24, find AB.

Q14:

Given that Aijk=35+, Bijk=533, Cijk=2+4, and (+𝑚)AB is perpendicular to the vector C, determine 𝑚.

  • A 1 5
  • B7
  • C15
  • D 7

Q15:

If Aijk=3+3+4𝑚, Bijk=467, and AB, find the value of 𝑚.

  • A 3 1 4
  • B 1 7 4
  • C 1 5 1 4
  • D 3 2

Q16:

Determine 𝑚, given that the scalar product of the two vectors Aijk=𝑚66 and Bijk=2+84 is 27.

Q17:

For the unit vectors i, j, k, what is kk?

Q18:

For the unit vectors i, j, k, what is jk?

Q19:

V and W are two vectors, where V=6,0,4 and W=0,2,1. Is VW? Justify your answer.

  • AYes because VW0.
  • BNo because VW0.
  • CNo because VW=0.
  • DYes because VW=0.

Q20:

If Aijk=253 and Bijk=42, find AB.

Q21:

Let V=5,1,2 and W=4,4,3. Calculate VW.

Q22:

If Aijk=3+23, Bijk=22, and Cijk=23, find 2[2(2+4)]CABCB.

  • A 2 4 4 + 2 4 6 + 1 1 8 i j k
  • B 2 4 8 2 4 8 1 2 4 i j k
  • C 2 4 6 + 2 4 7 + 1 2 1 i j k
  • D j k 2

Q23:

For the unit vectors i, j, k, what is ki?

Q24:

Given that 5=1015+10Aijk and 4=8+124Bijk, determine 44AB.

Q25:

Suppose A=1,2,7, ||=13B, and the angle between the two vector is 135. Find AB to the nearest hundredth.

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