The diagram shows two vectors, 𝐀 and 𝐁. What is the scalar product of 𝐀 and 𝐁?
In this question, we are presented with a diagram of two vectors and asked to find their scalar product. We see that we have a vector 𝐀 with a magnitude of four and that we have a vector 𝐁 with a magnitude of three. The angle 𝜃 between these two vectors is 90 degrees. Let’s recall that we can define the scalar product of two vectors 𝐀 and 𝐁 as the magnitude of vector 𝐀 multiplied by the magnitude of vector 𝐁 multiplied by the cos of the angle 𝜃 between them. So, let’s substitute in our values into this expression.
We get that the scalar product of 𝐀 and 𝐁 is equal to four. The magnitude of 𝐀 multiplied by three, the magnitude of 𝐁, multiplied by the cos of 90 degrees, which is the angle between 𝐀 and 𝐁. All that remains is to evaluate this expression. When we do this, we realize that the cos of 90 degrees gives us zero.
And so, the result of our calculation and the answer to the question of the scalar product of vectors 𝐀 and 𝐁 is zero. It’s worth pointing out that the fact that the cos of 90 degrees is zero means that since we multiply by the cos of the angle in our expression for the scalar product, then any two perpendicular vectors will have a scalar product of zero.