Lesson: Dot Product in 3D

In this lesson, we will learn how to find the dot product of two vectors and deduce whether they are perpendicular, parallel, or intersecting.

Sample Question Videos

  • 01:06
  • 01:50
  • 02:05

Worksheet: Dot Product in 3D • 25 Questions • 3 Videos

Q1:

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

Q2:

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

Q3:

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

Q4:

and Find .

Q5:

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

Q6:

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

Q7:

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

Q8:

Given that 𝐴 𝐵 𝐶 𝐷 is a square of side 33, determine 𝐴 𝐵 𝐶 𝐴 .

Q9:

Find .

Q10:

Given that 𝐴 = 5 𝑖 7 𝑗 + 7 𝑘 and 𝐵 = 7 𝑖 2 𝑗 5 𝑘 , determine 𝐴 𝐵 .

Q11:

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

Q12:

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

Q13:

If 𝑉 = 3 𝑖 2 𝑗 𝑘 and 𝑊 = 6 𝑖 + 4 𝑗 + 2 𝑘 , calculate 𝑉 𝑊 .

Q14:

If 𝐴 = ( 0 , 3 , 1 ) and 𝐵 = 2 𝑖 𝑗 4 𝑘 , find 𝐴 𝐵 .

Q15:

Given that 𝐴 = 3 𝑖 5 𝑗 + 𝑘 , 𝐵 = 5 𝑖 3 𝑗 3 𝑘 , 𝐶 = 2 𝑖 𝑗 + 4 𝑘 , and 𝐴 + 𝑚 𝐵 is perpendicular to the vector 𝐶 , determine 𝑚 .

Q16:

If 𝐴 = 3 𝑖 + 3 𝑗 + 4 𝑚 𝑘 , 𝐵 = 4 𝑖 6 𝑗 7 𝑘 , and 𝐴 𝐵 , find the value of 𝑚 .

Q17:

Determine 𝑚 , given that the scalar product of the two vectors 𝐴 = 𝑚 𝑖 6 𝑗 6 𝑘 and 𝐵 = 2 𝑖 + 8 𝑗 4 𝑘 is 27.

Q18:

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

Q19:

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

Q20:

and are two vectors, where and . Is ? Justify your answer.

Q21:

If 𝐴 = 2 𝑖 5 𝑗 3 𝑘 and 𝐵 = 4 𝑖 2 𝑗 𝑘 , find 𝐴 𝐵 .

Q22:

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

Q23:

If 𝐴 = 3 𝑖 + 2 𝑗 3 𝑘 , 𝐵 = 2 𝑖 2 𝑗 𝑘 , and 𝐶 = 2 𝑖 𝑗 3 𝑘 , find 2 𝐶 2 𝐴 2 𝐵 + 4 𝐶 𝐵 .

Q24:

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

Q25:

Given that 5 𝐴 = 1 0 𝑖 1 5 𝑗 + 1 0 𝑘 and 4 𝐵 = 8 𝑖 + 1 2 𝑗 4 𝑘 , determine 4 𝐴 4 𝐵 .

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