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Vector Multiplied by a Scalar

Multiplication by a constant which affects magnitude.

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Vocabulary

Complete the chart.
Word Definition
Vector ____________________________________________________________
____________ the length of a line segment or vector
Scalar ____________________________________________________________
____________ To move a vector on a coordinate system without changing its length or orientation

Multiply by a Scalar

A vector can be multiplied by a real number. This real number is called a _____________.

The product of a vector \begin{align*}\vec{a}\end{align*} and a scalar \begin{align*}k\end{align*} is a vector, written \begin{align*}\vec{ka}\end{align*} . It has the same direction as _____ with a magnitude of ________ if . If \begin{align*}k < 0\end{align*} , the vector has the opposite direction of _____ and a magnitude of ________ .


\begin{align*}\vec{i}\end{align*} is in standard position with terminal point (3, 7) and \begin{align*}\vec{j}\end{align*} is in standard position with terminal point (-2, 4).

  1. Find the coordinates of the terminal point of \begin{align*}5\vec{i} - \vec{j}\end{align*} .
  2. Find the coordinates of the terminal point of \begin{align*}0.2\vec{i} - 0.7\vec{j}\end{align*} .
  3. Find the coordinates of the terminal point of \begin{align*}8\vec{i} + 1.1\vec{j}\end{align*} .

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Vectors with the same magnitude and direction are ____________. This means that the same ordered pair could represent many different vectors.

Think back: How do you find the slope of a line? _______________________

Are these two vectors equal? 

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In each question below, the initial and terminal coordinates for a vector are given. If the vector is translated so that it is in standard position (with the initial point at the origin), what are the new terminal coordinates?

  1. initial (1, 6) and terminal (4, -2)
  2. initial (5, 3) and terminal (1, -6)
  3. initial (8, 4) and terminal (-9, 7)


Find the slope of each vector below with the given terminal coordinates. Assume the vector is in standard position.

  1. terminal (4, 9)
  2. terminal (7, 5)
  3. terminal (-3, 2)

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