In this video, we will discuss dot products, orthogonality, and vector components. Let A and B be vectors with entries A1 through An and B1 through Bn. The dot product of A and B, sometimes called the scalar product of A and B, is algebraically defined as the scalar obtained by multiplying corresponding entries in summing the result. The dot product of A and B also has a geometric definition. If theta is the angle between A and B, then the dot product of A and B is the length of A times the length of B times cosine of the angle theta. We can use either definition to compute the dot product of two vectors, and the result will always be a scalar. Let's look at an example. Consider the vectors U and V given. Using the algebraic definition of the dot product, we compute the dot product of U and V is 4. We can leverage this fact and conjunction with the geometric definition of the dot product to compute the angle between the vectors U and V. By computing the lengths of U and V, we have an equation where the only unknown is theta, the angle between U and V, which we can solve for. Next, we will discuss orthogonality. Two vectors are called orthogonal, if their dot product is 0. Using the geometric definition of the dot product, we see that the vectors A and B are orthogonal, if and only if, one or both of their lengths are 0, or the angle between them is 90 degrees. Consider the vectors U and V from earlier in the new vector W. U dot V equals 4, so U and V are not orthogonal. But U dot W is 0, therefore U and W are orthogonal. Lastly, we will discuss the component of one vector in the direction of another. The component of the vector A in the direction of a non-zero vector B is the unique vector in the direction of B, so that when you subtract off the component of the vector A in the direction of B from A, it becomes orthogonal to B. Let's look at that with a picture. The component of A in the direction B is a vector in the direction of B, and when you take A minus that component, it's orthogonal to B. Using these two properties, we can develop a formula for computing the component of the vector A in the direction B. It is worth noting that some other courses use the word projection instead of component. Let's look at an example with our familiar vectors U and V. The component of U in the direction of V can be computed with the formula. And after evaluating all dot products, we see it is 4, 14 times the vector V.