The scalar (aka dot) product, and its connection to angles between vectors.
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 measures the extent to which two vectors point in the same direction. The bigger the vectors and the more they point in the same direction, the bigger the scalar product, since only the "parallel parts" contribute to the dot product. It's is a scalar quantity measuring the amount of similarity.
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:
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°.
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:
Nice work completing Scalar product, here's a quick recap of what we covered:
Exercises checked off
The scalar (aka dot) product, and its connection to angles between vectors.
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 measures the extent to which two vectors point in the same direction. The bigger the vectors and the more they point in the same direction, the bigger the scalar product, since only the "parallel parts" contribute to the dot product. It's is a scalar quantity measuring the amount of similarity.
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:
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°.
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:
Nice work completing Scalar product, here's a quick recap of what we covered:
Exercises checked off