To find the component of vector in the direction of vector , divide their dot product by the magnitude of :
To find the component of vector in the direction of vector , divide their dot product by the magnitude of :
This is the scalar component, so it is a signed length. If the question requires a vector, calculate the vector projection instead.
The unit vector in the direction of is
The dot product of with this unit vector measures how much of acts in that direction:
The sign provides contextual information. A positive answer means the component acts in the same direction as ; a negative answer means it acts in the opposite direction. A zero component means the vectors are perpendicular.
| Required quantity | Formula and meaning |
|---|---|
| Scalar component | , a signed length |
| Vector component | , a vector parallel to |
Let and . First calculate
Therefore, the scalar component is
For the vector projection, multiply this scalar component by the unit vector in the required direction:
The resulting vector is parallel to , while its magnitude equals the absolute value of the scalar component. The remaining perpendicular component could be found by subtracting the projection from .
A common misconception is dividing by . The denominator must use the vector defining the direction: here, . Another mistake is reporting a scalar when the question asks for a vector.
This is AHL 3.13: Scalar and vector products, so it is HL-only content. State whether you are finding a scalar component or vector projection, show the dot product and magnitude explicitly, and give the final answer exactly or to three significant figures. On a GDC-active paper, calculator output does not replace the mathematical setup.