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.