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:

𝐴𝐵𝐶 is a right 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⟩+𝑡⟨−10,2,−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=−5−2+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:

𝐴𝐵𝐶𝐷 is a trapezoid, where 𝐴𝐷∥𝐵𝐶, 𝑚∠𝐴=𝑚∠𝐵=90∘, 𝑚∠𝐶=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:

𝐴𝐵𝐶 is a right-angled triangle in which 𝐴𝐵=28cm, 𝐵𝐶=21cm and 𝑚∠𝐴𝐵𝐶=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−18√3
  • C54
  • D−27√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.

  • A52
  • B152
  • C75√32
  • D15√32
  • E5√32

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||×𝜃ABcos
  • B||𝜃Bcos
  • C||𝜃Acos
  • D||×𝜃ABcos
  • E(⋅)𝜃ABcos

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.

Q19:

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

  • AAijk=8.72+3.35+3.58
  • BAijk=1.28+3.35+9.34
  • CAijk=3.58+9.34+9.34
  • DAijk=3.35+8.72+3.58

Q20:

Given that A=⟨−6,3,−2⟩ and B=⟨4,1,6⟩, determine the component of A along B.

  • A67
  • B−337
  • C−33√5353
  • D−5√5353

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−10
  • B75
  • C65
  • D10
  • E−65

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