The diagram shows two vectors, 𝐀 and 𝐁. What is their scalar product?
This question presents us with a diagram and asks us to find the scalar product of the two vectors in that diagram. We can see from the diagram that we have a vector 𝐀 with a magnitude of 6.4 and that we have a vector 𝐁 with a magnitude of 3.2. 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.
The diagram gave us values for the magnitude of 𝐀, the magnitude of 𝐁, and the value of 𝜃. Now we just need to substitute in these values into the expression for the scalar product. Doing this, we get that the scalar product of 𝐀 and 𝐁 is equal to 6.4, the magnitude of 𝐀, multiplied by 3.2, the magnitude of 𝐁, multiplied by the cos of 90 degrees, the angle 𝜃 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. We note that the fact that the cos of 90 degrees is zero means that since we multiply by the cosine of the angle in our expression for the scalar product, then any two perpendicular vectors will have a scalar product of zero.