Practice Dot Product in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The dot product of two vectors \mathbf{a} = \langle a_1, a_2 \rangle and \mathbf{b} = \langle b_1, b_2 \rangle is the scalar \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2. Equivalently, \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta, where \theta is the angle between the vectors.
The dot product measures how much two vectors point in the same direction. If they point the same way, the dot product is large and positive. If perpendicular, it is zero. If they point in opposite directions, it is negative. Think of it as a 'similarity score' for directions.
Example 1
easyFind \langle 1, 2 \rangle \cdot \langle 3, -1 \rangle.
Example 2
mediumFind the angle between \mathbf{a} = \langle 1, 0 \rangle and \mathbf{b} = \langle 1, 1 \rangle.
Example 3
easyAre \langle 2, 3 \rangle and \langle -3, 2 \rangle perpendicular?
Example 4
hardFind \langle 2, -1, 3 \rangle \cdot \langle 4, 5, -2 \rangle.