Components of Vectors | American School

Here is a comprehensive set of 100 practice problems on finding the components of a vector (primarily in 2D, using the standard x-y coordinate system).

The main types include:

Given magnitude and direction angle θ (measured counterclockwise from the positive x-axis).
Given initial and terminal points.
A few mixed or reverse problems (finding magnitude/direction from components, or resolving into components).

Use these formulas (with angles in degrees unless specified):

x-component (horizontal): $A_x = |A| \cos \theta$ $A$ $x$ $$ $=$ $∣$ $A$ $∣$ $cos$ $θ$
y-component (vertical): $A_y = |A| \sin \theta$ $A$ $y$ $$ $=$ $∣$ $A$ $∣$ $sin$ $θ$
Vector in component form: $\vec{A} = \langle A_x, A_y \rangle$ $A$ $=$ $⟨$ $A$ $x$ $$ $,$ $A$ $y$ $$ $⟩$ or $A_x \hat{i} + A_y \hat{j}$ $A$ $x$ $$ $i$ $^$ $+$ $A$ $y$ $$ $j$ $^$ $$

For position vectors from point P(x₁, y₁) to Q(x₂, y₂):
$\vec{PQ} = \langle x_2 - x_1, y_2 - y_1 \rangle$

$P$ $Q$

$$ $=$ $⟨$ $x$ $2$ $$ $-$ $x$ $1$ $$ $,$ $y$ $2$ $$ $-$ $y$ $1$ $$ $⟩$

Round answers to 2 decimal places where appropriate (exact values like √3/2 are encouraged when possible). Solutions are provided after each problem for self-checking.

Problems 1–20: Magnitude and Angle (Basic)

Magnitude 10, θ = 0°
Magnitude 5, θ = 90°
Magnitude 8, θ = 180°
Magnitude 12, θ = 270°
Magnitude 20, θ = 30°
Magnitude 15, θ = 45°
Magnitude 25, θ = 60°
Magnitude 10, θ = 120°
Magnitude 18, θ = 135°
Magnitude 7, θ = 150°
Magnitude 14, θ = 210°
Magnitude 9, θ = 225°
Magnitude 16, θ = 240°
Magnitude 11, θ = 300°
Magnitude 22, θ = 315°
Magnitude 13, θ = 330°
Magnitude 6, θ = 45° (express exactly)
Magnitude 4√2, θ = 135°
Magnitude 50, θ = 53° (approx.)
Magnitude 100, θ = 37°

Problems 21–40: Magnitude and Angle (Intermediate)

Magnitude 30, θ = 25°
Magnitude 40, θ = 65°
Magnitude 28, θ = 110°
Magnitude 35, θ = 155°
Magnitude 42, θ = 200°
Magnitude 19, θ = 250°
Magnitude 26, θ = 280°
Magnitude 33, θ = 340°
Magnitude 17, θ = 15°
Magnitude 24, θ = 75°
Magnitude 31, θ = 105°
Magnitude 39, θ = 165°
Magnitude 46, θ = 195°
Magnitude 12.5, θ = 285°
Magnitude 8.5, θ = 345°
Magnitude 55, θ = 10°
Magnitude 67, θ = 80°
Magnitude 23, θ = 220°
Magnitude 29, θ = 310°
Magnitude 41, θ = 350°

Problems 41–60: Initial and Terminal Points

From (0,0) to (5,12)
From (3,4) to (8,9)
From (-2,1) to (4,7)
From (1,-3) to (-5,2)
From (0,5) to (12,0)
From (-4,-6) to (3,5)
From (2,2) to (2,-8)
From (7,-1) to (-3,4)
From (0,0) to (-9,12)
From (10,15) to (1,1)
From (-5,8) to (6,-3)
From (4,0) to (4,10)
From (-1,-1) to (9,9)
From (6,7) to (-2,-4)
From (0,0) to (20,21)
From (3,-2) to (-7,11)
From (-8,5) to (4,-5)
From (15,0) to (0,8)
From (1,12) to (13,4)
From (-10,-10) to (5,15)

Problems 61–80: Magnitude and Angle (Advanced/Mixed Angles)

Magnitude 50, θ = 120°
Magnitude 36, θ = 240°
Magnitude 48, θ = 300°
Magnitude 27, θ = 22.5°
Magnitude 64, θ = 157.5°
Magnitude 18, θ = 202.5°
Magnitude 32, θ = 337.5°
Magnitude 21, θ = 48°
Magnitude 44, θ = 132°
Magnitude 29, θ = 228°
Magnitude 38, θ = 312°
Magnitude 52, θ = 18°
Magnitude 61, θ = 72°
Magnitude 73, θ = 108°
Magnitude 85, θ = 162°
Magnitude 14, θ = 198°
Magnitude 33, θ = 252°
Magnitude 47, θ = 288°
Magnitude 59, θ = 342°
Magnitude 66, θ = 5°

Problems 81–100: Mixed (Including Reverse and 3D Hints)

81–90: Given components, find magnitude and direction θ (for verification practice).
81. ⟨3, 4⟩
82. ⟨-5, 12⟩
83. ⟨8, -15⟩
84. ⟨-7, -24⟩
85. ⟨20, 21⟩
86. ⟨-9, 40⟩
87. ⟨12, 5⟩
88. ⟨-16, -30⟩
89. ⟨0, 25⟩
90. ⟨-18, 0⟩

91–95: Resolve into components (same as 1–20 style but with decimals).
91. Magnitude 7.5, θ = 36.87°
92. Magnitude 12.8, θ = 53.13°
93. Magnitude 25.6, θ = 143.13°
94. Magnitude 18.4, θ = 216.87°
95. Magnitude 9.2, θ = 323.13°

96–100: Position vectors or slight variations.
96. From (2,3) to (10,15)
97. From (-4,7) to (8,-2)
98. Magnitude 40 at 225° (exact form preferred)
99. Magnitude 100 at 0° + a small y-component of 10 (find adjusted x)
100. A vector of length 50 at 60° north of east (treat as θ=30° from x if east is +x, north +y)