How to Calculate a Vector Dot Product

Vectors are mathematical objects having a size, or magnitude, and a direction. Vectors are useful in physics when describing directional quantities such as velocity, acceleration and force.The dot product of two vectors, also known as their scalar product, is a way of multiplying vectors, arriving at a scalar quantity (in other words having a magnitude but no direction).Vector dot products are defined in general for any number of dimensions.

Things You'll Need

• Pen and paper

Instructions

Calculating the dot product, or scalar product, of two vectors

1. Choose if you wish to use trigonometry or algebra. If you know the magnitude of the two vectors and the angle between them, use trigonometry by going to step 2. If you know the components of the vectors, use algebra by going to step 3.

2. Multiply the magnitude of the two vectors by the cosine of the angle theta between them: |X| * |Y| * cos(theta). This is the dot product of the two vectors.

3. Multiply the first components of the two vectors by each other:x_1 * y_1.

4. Multiply the second components of the two vectors by each other: x_2 * y_2and add the product of these to the product of the first components. Repeat this step with the third components, fourth and so on. The overall sum will thus be: (x_1 * y_1) + (x_2 * y_2) + (x_3 * y_3) + ... + (x_n * y_n), where n is the number of dimensions of the two vectors. This overall sum is the dot product of the two vectors.

Tips & Warnings

• The two methods can be shown to be equivalent to each other, and if both are carried out correctly, the results will be the same. You should choose the method that is easier given the information you have in hand when you start out.

• The geometric interpretation of the dot product is the component of one vector along the direction of the other, multiplied by the magnitude of that other vector. If the second vector has a magnitude of 1, the dot product is simply the component of the first vector along the direction of the second, unit, vector.

• Note that the two vectors must be in the same space. This means that they should have the same number of components, and that the components are based on the same Cartesian coordinate system ( X, Y, Z). If you have one vector given in Cartesian coordinates and another in spherical coordinates (R, theta, phi) you will need to convert the latter into Cartesian coordinates before using the algebraic method.

• When calculating the dot product of 4-vectors in space-time in physics, the zeroth component (the time-like one) has the opposite sign compared to the space-like components.

Resources

• Wolfram Mathworld