Consider the vector in the given diagram. What are the coordinates of its terminal point? What are the coordinates of its initial point? What are the components of the vector?
We know that the terminal point of any vector is its endpoint. Any coordinate pair has an 𝑥-value and a 𝑦-value. We go along the 𝑥-axis to find the first value and then up or down the 𝑦-axis to find the second value. To reach the endpoint of the vector, we go along the 𝑥-axis to negative one and down the 𝑦-axis to negative two. Therefore, the coordinates of the terminal point of vector 𝐯 are negative one, negative two.
The initial point of any vector is its start point. In this question, this is situated at two, negative two. The coordinates of the initial point of vector 𝐯 are two, negative two.
The final part of our question asks us to work out the components of vector 𝐯. We can do this by subtracting the initial point from the terminal point. The 𝑥-component of the vector will therefore be equal to negative one minus two. The 𝑦-component will be equal to negative two minus negative two. This is the same as negative two plus two. Negative one minus two is equal to negative three. And negative two plus two equals zero. The components of vector 𝐯 are negative three, zero.
This answer leads us to a general rule. Any horizontal vector will have a 𝑦-component equal to zero. The vector will be of the form 𝑥, zero, as there is no change in the 𝑦-coordinate or component from the initial to the terminal point. Likewise, any vertical vector will have an 𝑥-component equal to zero. Our 𝑥-component in this question is negative as the vector is moving to the left. Had the vector moved to the right, the 𝑥-component would be positive. In this question, we moved three unit squares to the left. Therefore, the 𝑥-component is equal to negative three.