Vector Magnitude — Worksheet (with step-by-step
solutions)
Instructions:
Work through the questions in the left column. Solutions are provided below for checking your work. This
sheet is printable.
Questions
1. Find the magnitude of the vector a = 6i + 8j .
2. Find the magnitude of the vector b = −7i + 24j .
3. If c = xi + 12j has magnitude 13, find x .
4. Find the magnitude of the vector p = 2i − 3j + 6k .
5. A vector q = xi + yj + zk has magnitude 7. If x = 3 and y = 4 , find z .
6. Verify the magnitude of r = −i + 2j − 2k .
7. A ship sails 4 km east and then 3 km north. Represent this as a vector and find the straight-line
displacement.
8. A person walks 5 km north then 12 km east. Find the straight-line distance from the starting point.
9. Find the magnitude of the force vector F = 10i + 6j (in Newtons).
10. Two perpendicular unit vectors are added. Find the magnitude of the resultant.
Step-by-step Solutions
1. a = 6i + 8j .
∣a∣ =
62 + 82 =
36 + 64 =
100 = 10.
2. b = −7i + 24j .
∣b∣ =
(−7)2 + 242 =
49 + 576 =
625 = 25.
3. c = xi + 12j, ∣c∣ = 13.
x2 + 122 = 132 ⟹ x2 + 144 = 169 ⟹ x2 = 25 ⟹ x = ±5.
4. p = 2i − 3j + 6k .
∣p∣ =
22 + (−3)2 + 62 =
5. q = 3i + 4j + zk, ∣q ∣ = 7.
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4 + 9 + 36 =
49 = 7.
32 + 42 + z 2 = 72 ⟹ 9 + 16 + z 2 = 49 ⟹ z 2 = 24
impliesz = ±2 6.
6. r = −i + 2j − 2k .
∣r∣ =
(−1)2 + 22 + (−2)2 =
1+4+4=
9 = 3.
7. Displacement vector = (4, 3) = 4i + 3j .
Distance = 42 + 32 = 16 + 9 = 25 = 5 km.
8. Displacement vector = (12, 5) = 12i + 5j .
Distance = 122 + 52 = 144 + 25 = 169 = 13 km.
9. F = 10i + 6j .
∣F ∣ =
102 + 62 =
100 + 36 =
136 = 2 34 ≈ 11.66 N.
10. Two perpendicular unit vectors each have magnitude 1.
Resultant magnitude = 12 + 12 = 2.
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