Worksheet: Vector Projection

In this worksheet, we will practice finding the projection of a vector onto another.

Q1:

The cube shown has sides of length 4 4 1 7 . Find the scalar projection of 𝑂 𝐴 onto 𝐶 𝐵 , giving your answer correct to two decimal places.

Q2:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 , where 𝐴 𝐵 = 1 7 c m , 𝐵 𝐶 = 1 1 c m , and 𝐷 is the midpoint of 𝐴 𝐶 . Find the algebraic projection of 𝐴 𝐷 in the direction of 𝐶 𝐵 .

Q3:

Given that | | = 1 2 A , | | = 2 7 B , and the measure of the angle between A and B is 6 0 , determine the algebraic projection of ( + ) A B in the direction of 𝐵 .

Q4:

Determine, to the nearest hundredth, the length of the projection of the vector passing through the points ( 1 0 , 4 , 6 ) and ( 4 , 5 , 9 ) to the straight line 𝑟 = ( 4 , 6 , 0 ) + 𝑡 ( 1 0 , 2 , 1 0 ) .

Q5:

The cube shown has sides of length 8 3 . Find the scalar projection of 𝑂 𝐴 onto 𝐶 𝐵 , giving your answer correct to two decimal places.

Q6:

The cube shown has sides of length 30. Find the scalar projection of 𝑂 𝐴 onto 𝐶 𝐵 , giving your answer correct to two decimal places.

Q7:

The cube shown has sides of length 2 8 3 . Find the scalar projection of 𝑂 𝐴 onto 𝐶 𝐵 , giving your answer correct to two decimal places.

Q8:

The cube shown has sides of length 25. Find the scalar projection of 𝑂 𝐴 onto 𝐶 𝐵 , giving your answer correct to two decimal places.

Q9:

Consider the points 𝐴 ( 5 , 1 , 8 ) and 𝐵 ( 3 , 9 , 6 ) . Find the component of the vector 𝑉 = 5 𝑖 2 𝑗 + 2 𝑘 in the direction 𝐴 𝐵 rounded to the nearest hundredth.

Q10:

Given that is a square having a side length of 53 cm, calculate the algebraic projection of in the direction of .

Q11:

𝐴 𝐵 𝐶 𝐷 is a trapezium, where 𝐴 𝐷 𝐵 𝐶 , 𝑚 𝐴 = 𝑚 𝐵 = 9 0 , 𝑚 𝐶 = 6 0 , 𝐴 𝐷 = 6 0 c m , and 𝐵 𝐶 = 1 0 1 c m . Determine the algebraic projection of 𝐶 𝐷 on the direction of 𝐶 𝐵 .

Q12:

Determine whether the following is true or false: If the component of a vector in the direction of another vector is zero, then the two are perpendicular.

  • Atrue
  • Bfalse

Q13:

is a right-angled triangle in which , and . Determine the algebraic projection of in the direction of .

  • A22.4
  • B
  • C
  • D28

Q14:

Given that the measure of the smaller angle between A and B is 1 5 0 , and | | = 5 4 B , determine the component of vector B along A .

  • A 1 8 3
  • B27
  • C54
  • D 2 7 3

Q15:

If | | = 5 A , | | | | = 1 5 B , and the measure of the angle between them is 3 0 , find the algebraic projection of B in the direction of 𝐴 .

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

Q16:

Determine, to the nearest hundredth, the component of vector 𝑉 along 𝐴 𝐵 , given that 𝑉 = ( 7 , 2 , 1 0 ) and the coordinates of 𝐴 and 𝐵 are ( 1 , 4 , 8 ) and ( 3 , 2 , 0 ) , respectively.

Q17:

Find the component of vector 𝐴 in the direction of 𝐵 , where 𝜃 is the included angle between them.

  • A 𝐵 𝜃 c o s
  • B 𝐴 𝐵 𝜃 c o s
  • C 𝐴 × 𝐵 𝜃 c o s
  • D 𝐴 𝜃 c o s
  • E 𝐴 × 𝐵 𝜃 c o s

Q18:

Determine the magnitude of the component of 𝐹 = 2 3 𝑖 + 1 7 𝑗 in the direction of 𝐴 𝐵 , given that the coordinates of the points 𝐴 and 𝐵 are ( 2 , 3 ) and ( 6 , 6 ) respectively.

  • A0.2
  • B27.4
  • C8.2
  • D28.6

Q19:

The vector 𝑂 𝐴 has length 10. Write this vector using i , j , and k , with components accurate to two decimal places.

  • A A i j k = 1 . 2 8 + 3 . 3 5 + 9 . 3 4
  • B A i j k = 3 . 5 8 + 9 . 3 4 + 9 . 3 4
  • C A i j k = 8 . 7 2 + 3 . 3 5 + 3 . 5 8
  • D A i j k = 3 . 3 5 + 8 . 7 2 + 3 . 5 8

Q20:

Given that 𝐴 = ( 6 , 3 , 2 ) and 𝐵 = ( 4 , 1 , 6 ) , determine the component of 𝐴 along 𝐵 .

  • A 5 5 3 5 3
  • B 3 3 7
  • C 6 7
  • D 3 3 5 3 5 3

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