Video: Multiplying a 2D Vector by a Scalar Graphically

Video Transcript

Vector 𝐀 is represented by the following graph. Which of the following graphs represents negative two 𝐀?

Vector 𝐀 goes from the origin to the point one, one. This means that it has an 𝑥-component of one and a 𝑦-component of one. Vector 𝐀 is equal to one, one. We want to multiply vector 𝐀 by negative two. We recall that when multiplying a vector by a scalar, we need to multiply each of the individual components by the scalar. Multiplying negative two by one gives us negative two. Therefore, the vector that corresponds to negative two 𝐀 is negative two, negative two.

Let’s now consider the five options we are given and what vectors they represent. Graph (A) goes from the origin to negative two, negative two. This means it corresponds to the vector negative two, negative two. This is the same as negative two 𝐀, which suggests this is the correct graph.

Graph (B) shows the vector one, negative two. Graph (C) shows the vector one, two. Graph (D) shows the vector one, 0.5. And graph (E) shows the vector 0.5, 0.5. This confirms that graph (A) does indeed represent negative two 𝐀. This leads us to a key rule when multiplying by negative scalars. When we multiply any vector by a negative scalar other than negative one, the vector will change direction and magnitude. Multiplying a vector by negative two, as in this case, will double the magnitude, and the direction of the vector will be the opposite of the original direction. This can be shown on the coordinate plane by the green arrow.