FAQs on How to Calculate the Dot Product of 3D Vectors

Question

Accepted Answer

Q. What is the dot product in 3D vectors?

A. The dot product in 3D vectors is a mathematical operation used to determine the geometric relationship between two three-dimensional vectors. The dot product can be thought of as the “dot” between the two vectors, referred to as v and w, which produces a scalar value. This scalar value describes how closely aligned or orthogonal (perpendicular) these two vectors are in relation to each other.

Q. How do you calculate the dot product of 3D Vectors?

A. In order to calculate the dot product, we must have both xyz components for each vector represented: v = [v1, v2, v3] and w = [w1, w2, w3] . The dot product formula looks like this: v•w=v1*w1 + v2*w2 + v3*w3 , which will result in a single numerical value describing their relationship. To complete this calculation with real numbers we would rewrite it like this (using an example of 1 point in space): [5,7,-4] • [-8,-6,9] = 5(-8)+7(-6)+(-4)(9) = -52 + 42 – 36 = -46 . Thus, the answer to our equation is -46 . As you can see from the equation itself; when two vectors are orthogonal ( perpendicular ) to one another their individual components cancel out upon multiplication resulting in zero for the entire equation’s sum—regardless of their magnitudes or lengths.

Q. Is there an easier way of calculating a Dot Product?

A. Yes! There is also an alternate method for calculating a dot product that doesn’t involve breaking down each vector’s individual components or multiplying each component by its respective counterpart on the other vector: known as the magnitude-angle method.. This method works by using either cosine law or Pythagorean Theorem to solve for angles and lengths separately before actually calculating any products within your equation and results in much fewer steps overall than traditionally stated above!