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 4417. Find the scalar projection of 𝑂𝐴 onto 𝐶𝐵, giving your answer correct to two decimal places.

Q2:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵, where 𝐴𝐵=17cm, 𝐵𝐶=11cm, and 𝐷 is the midpoint of 𝐴𝐶. Find the algebraic projection of 𝐴𝐷 in the direction of 𝐶𝐵.

Q3:

Given that ||=12A, ||=27B, and the measure of the angle between A and B is 60, determine the algebraic projection of (+)AB in the direction of 𝐵.

Q4:

Determine, to the nearest hundredth, the length of the projection of the vector passing through the points (10,4,6) and (4,5,9) to the straight line r=4,6,0+𝑡10,2,10.

Q5:

The cube shown has sides of length 83. Find the scalar projection of 𝑂𝐴 onto 𝐶𝐵, giving your answer correct to two decimal places.

Q6:

The cube shown has sides of length 30. Find the scalar projection of 𝑂𝐴 onto 𝐶𝐵, giving your answer correct to two decimal places.

Q7:

The cube shown has sides of length 283. Find the scalar projection of 𝑂𝐴 onto 𝐶𝐵, giving your answer correct to two decimal places.

Q8:

The cube shown has sides of length 25. Find the scalar projection of 𝑂𝐴 onto 𝐶𝐵, 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 Vijk=52+2 in the direction 𝐴𝐵 rounded to the nearest hundredth.

Q10:

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

Q11:

𝐴 𝐵 𝐶 𝐷 is a trapezium, where 𝐴𝐷𝐵𝐶, 𝑚𝐴=𝑚𝐵=90, 𝑚𝐶=60, 𝐴𝐷=60cm, and 𝐵𝐶=101cm. Determine the algebraic projection of 𝐶𝐷 on the direction of 𝐶𝐵.

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.

  • Afalse
  • Btrue

Q13:

𝐴 𝐵 𝐶 is a right-angled triangle in which 𝐴𝐵=28cm, 𝐵𝐶=21cm and 𝑚𝐴𝐵𝐶=90. Determine the algebraic projection of 𝐶𝐴 in the direction of 𝐵𝐴, rounding your answer to two decimal places.

Q14:

Given that the measure of the smaller angle between A and B is 150, and ||=54B, determine the component of vector B along A.

  • A27
  • B 1 8 3
  • C54
  • D 2 7 3

Q15:

If ||=5A, ||||=15B, and the measure of the angle between them is 30, find the algebraic projection of B in the direction of 𝐴.

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

Q16:

Determine, to the nearest hundredth, the component of vector V along 𝐴𝐵, given that V=7,2,10 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 | | × 𝜃 A B c o s
  • B | | 𝜃 B c o s
  • C | | 𝜃 A c o s
  • D | | × 𝜃 A B c o s
  • E ( ) 𝜃 A B c o s

Q18:

Determine the magnitude of the component of Fij=23+17 in the direction of 𝐴𝐵, given that the coordinates of the points 𝐴 and 𝐵 are (2,3) and (6,6) respectively.

  • A8.2
  • B27.4
  • C28.6
  • D0.2

Q19:

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

  • A A i j k = 8 . 7 2 + 3 . 3 5 + 3 . 5 8
  • B A i j k = 1 . 2 8 + 3 . 3 5 + 9 . 3 4
  • C A i j k = 3 . 5 8 + 9 . 3 4 + 9 . 3 4
  • 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 6 7
  • B 3 3 7
  • C 3 3 5 3 5 3
  • D 5 5 3 5 3

Q21:

Given that 𝐴𝐵𝐶𝐷 is a rectangle in which 𝐴𝐵=65cm and 𝐵𝐶=10cm, define the algebraic projection of 𝐵𝐷 in the direction of 𝐵𝐶.

  • A 1 0
  • B75
  • C65
  • D10
  • E 6 5

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