Introduction to and applications of vector and scalar products
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Practice exam-style scalar and vector products problems
The scalar product, also called the dot product, takes two vectors and produces a scalar (a number). For two vectors v=⎝⎛v1v2v3⎠⎞ and w=⎝⎛w1w2w3⎠⎞, the scalar product is calculated as:
This operation combines corresponding components of each vector, resulting in a single numerical value.
The scalar product also has a geometric interpretation involving the angle θ between two vectors:
Equivalently, isolating cosθ:
The angle θ is measured between the heads of v and w:
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If the scalar product of two vectors is negative, then
and thus θ must be an obtuse angle: 90°<θ≤180°.
But since vectors always form an acute AND an obtuse angle, we can find the acute angle by subtracting
whenever θ>90°.
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Since v⋅w=∣v∣∣w∣cosθ, if two vectors are perpendicular then cos90°=0⇒ their scalar product is 0.
For any vectors u, v, and w, and scalar k:
The vector product, sometimes called the cross product, of two vectors v and w is given by:
This vector is perpendicular to both original vectors. Using the right-hand rule, if your index finger points in the direction of v and your middle finger points towards w, your thumb points towards v×w. The vector product thus creates a new vector perpendicular to both original vectors.
The magnitude of the vector product v×w gives the area of the parallelogram formed by vectors v and w:
Intuitively, this happens because the magnitude combines both vectors’ lengths and how "spread out" they are from each other, capturing exactly the amount of two-dimensional space they span.
The magnitude of the vector product is connected to the sine of the angle between the vectors by the formula
Here, θ is the angle between vectors v and w. This relationship holds because the area of the parallelogram formed by the two vectors depends on their lengths and the angle separating them.
Specifically, the area is largest when the vectors are perpendicular (sin90∘=1) and zero when they are parallel (sin0∘=0).
For two vectors a and b, the magnitude of the component of vector a that acts in the direction of vector b is given by
The magnitude of the component of vector a that acts perpendicular to vector b, in the plane formed by the two vectors, is given by
where θ is the angle between the two vectors.
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For any vectors u, v, and w, and scalar k: