Resolve into Components, Divide by the Magnitude for a Unit Vector
Components, unit vectors and direction cosines all come from the same step: express a vector along perpendicular axes. From there, adding, projecting and scaling are simple arithmetic.
Vector Components and Unit Vectors is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
Components and angles
A vector of magnitude A at angle θ from the +x axis has components A cosθ and A sinθ, with the signs taken from the quadrant: at 240°, the components are -A/2 and -(sqrt 3)A/2. The angle with the y-axis is tan⁻¹(Aₓ / A_y), so for 2i + 3j it is tan⁻¹(2/3). Two vectors are parallel when their components are in the same ratio: (3, 6, 9) and (1, 2, 3) are parallel, but a single sign change breaks it.
Unit vectors, projections and direction cosines
A unit vector is the vector divided by its magnitude.
The direction cosines are l = Aₓ/|A|, m = A_y/|A|, n = A_z/|A|, and l² + m² + n² = 1: for i + j + sqrt(2) k the magnitude is 2, so they are (1/2, 1/2, 1/sqrt 2). For a bird flying at 20 m/s at 60° to the east axis and 60° to the vertical, l = 1/2 and n = 1/2, so m = 1/sqrt 2 and the velocity is 10i + 10 sqrt(2) j + 10k.
Finding the missing vector
To find the vector that must be added to others to reach a target, subtract the sum from the target. The sum of (i - 5j + 2k) and (3i + 6j - 7k) is 4i + j - 5k, so to reach the unit vector i the added vector is i - (4i + j - 5k) = -3i - j + 5k.
OlympIQ exam lens
- ✓Components: A cosθ, A sinθ with signs set by the quadrant
- ✓Unit vector = vector / magnitude; parallel vector of magnitude M = M x unit vector
- ✓Projection of a on n: (a · n̂) n̂
- ✓Direction cosines satisfy l² + m² + n² = 1
- ✓Missing vector = target - sum of the given vectors
Do not give the component of a along a direction as just the number a · n̂ when a vector is asked. The component along a direction is a vector, (a · n̂) n̂, whose magnitude is the dot product and whose direction is n̂.
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