2.1 Vectors in the plane  (Page 4/29)

Which of the following vectors are equivalent?

Vectors $\text{a},$ $\text{b},$ and $\text{e}$ are equivalent.

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We have seen how to plot a vector when we are given an initial point and a terminal point. However, because a vector can be placed anywhere in a plane, it may be easier to perform calculations with a vector when its initial point coincides with the origin. We call a vector with its initial point at the origin a standard-position vector    . Because the initial point of any vector in standard position is known to be $\left(0,0\right),$ we can describe the vector by looking at the coordinates of its terminal point. Thus, if vector v has its initial point at the origin and its terminal point at $\left(x,y\right),$ we write the vector in component form as

When a vector is written in component form like this, the scalars x and y are called the components of $\text{v}.$

Recall that vectors are named with lowercase letters in bold type or by drawing an arrow over their name. We have also learned that we can name a vector by its component form, with the coordinates of its terminal point in angle brackets. However, when writing the component form of a vector, it is important to distinguish between $⟨x,y⟩$ and $\left(x,y\right).$ The first ordered pair uses angle brackets to describe a vector, whereas the second uses parentheses to describe a point in a plane. The initial point of $⟨x,y⟩$ is $\left(0,0\right);$ the terminal point of $⟨x,y⟩$ is $\left(x,y\right).$

When we have a vector not already in standard position, we can determine its component form in one of two ways. We can use a geometric approach, in which we sketch the vector in the coordinate plane, and then sketch an equivalent standard-position vector. Alternatively, we can find it algebraically, using the coordinates of the initial point and the terminal point. To find it algebraically, we subtract the x -coordinate of the initial point from the x -coordinate of the terminal point to get the x component, and we subtract the y -coordinate of the initial point from the y -coordinate of the terminal point to get the y component.