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Scalar Multiplication
of Vectors

See how a single number can stretch, shrink, or flip a vector: visually on the coordinate plane and algebraically with \(c\langle v_x, v_y\rangle = \langle cv_x, cv_y\rangle\).

Precalculus

What Is a Vector?

A vector has both magnitude (length) and direction. In component form we write \(\vec{v} = \langle v_x,\, v_y \rangle\), where \(v_x\) is the horizontal part and \(v_y\) is the vertical part.

\(x\)
\(y\)
\(v_x = 3\)
\(v_y = 2\)
\(\vec{v} = \langle 3, 2 \rangle\)
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What Is a Scalar?

A scalar is a plain real number: it has size but no direction. When we multiply a vector by a scalar we scale the vector, which is where the word comes from.

Vector
Magnitude + direction: \(\langle 3, 2 \rangle\)
Scalar
Magnitude only: \(c = 2\), \(c = -3\), \(c = \tfrac{1}{2}\)

Why Does Scaling Matter?

Physics
Doubling a force vector: \(2\vec{F}\) doubles the push on an object.
Graphics
Zoom or resize objects by scaling their position vectors.
Linear Algebra
The span of a vector is the set of all its scalar multiples.

The Component-Wise Rule

1
Magnitude scales: \(\|c\,\vec{v}\| = |c|\;\|\vec{v}\|\).
2
Direction preserved when \(c \gt 0\); reversed when \(c \lt 0\).
3
Zero scalar: \(0 \cdot \vec{v} = \vec{0}\), the zero vector.

Positive Scalars Stretch

\(x\)
\(y\)
\(\vec{v}\)
\(2\vec{v}\)
\(3\vec{v}\)

Negative Scalars Reverse

\(x\)
\(y\)
\(\vec{v}\)
\(-\vec{v}\)
\(-2\vec{v}\)

Fractional Scalars Shrink

\(x\)
\(y\)
\(\vec{v} = \langle 4,3 \rangle\)
\(\tfrac{1}{2}\vec{v}\)

All Cases at a Glance

\(-2\vec{v}\)
\(-\vec{v}\)
\(\tfrac{1}{2}\vec{v}\)
\(\vec{v}\)
\(2\vec{v}\)

Scalar Multiplication vs. Dot Product

Scalar Multiplication Dot Product
Inputsscalar + vectorvector + vector
OutputA vectorA scalar
Formula\(c\langle v_x, v_y\rangle\)\(u_x v_x + u_y v_y\)
Direction?Same or reversedN/A

Step-by-Step Checklist

1
Identify the scalar \(c\) and the vector \(\vec{v} = \langle v_x, v_y \rangle\).
2
Multiply each component: \(c\vec{v} = \langle c\, v_x,\; c\, v_y \rangle\).
3
Determine direction: same if \(c \gt 0\), reversed if \(c \lt 0\).
4
Compute magnitude: \(\|c\vec{v}\| = |c|\,\|\vec{v}\|\).
5
Graph: draw from the origin to the new tip \((c\, v_x,\; c\, v_y)\).

Example 1: \(3\langle 1, 2\rangle\)

\(x\)
\(y\)
 1 2 3 1 2 3 4 5 6
\(\vec{v}\)
\(3\vec{v} = \langle 3,6 \rangle\)

Example 2: \(-2\langle 3, -1\rangle\)

\(x\)
\(y\)
\(\vec{v} = \langle 3,-1 \rangle\)
\(-2\vec{v} = \langle -6, 2 \rangle\)

Example 3: \(\tfrac{1}{2}\langle -4, 6\rangle\)

\(x\)
\(y\)
\(\vec{v}\)
\(\tfrac{1}{2}\vec{v}\)

Common Mistakes

✕ Adding Instead of Multiplying
Writing \(3\langle 2,5\rangle = \langle 5,8\rangle\) (adding 3) instead of \(\langle 6,15\rangle\) (multiplying).
✕ Forgetting to Flip
A negative scalar reverses direction. \(-2\langle 1,3\rangle = \langle -2,-6\rangle\), not \(\langle 2,6\rangle\).
✕ Scaling Magnitude Wrong
\(\|-3\vec{v}\| = 3\|\vec{v}\|\), not \(-3\|\vec{v}\|\). Take the absolute value of the scalar.

Summary Table

Scalar \(c\) Effect on Length Effect on Direction
\(c \gt 1\)StretchesSame
\(0 \lt c \lt 1\)ShrinksSame
\(c = 1\)No changeSame
\(c = 0\)Zero vectorNone
\(-1 \lt c \lt 0\)ShrinksReversed
\(c \lt -1\)StretchesReversed

Question 1 of 5

What is \(4\langle -1, 3\rangle\)?

Question 2 of 5

If \(\vec{v} = \langle 2, -5\rangle\), what is the magnitude of \(-3\vec{v}\)?

Question 3 of 5

A vector \(\vec{u}\) is multiplied by \(-\tfrac{1}{2}\). The resulting vector is:

Question 4 of 5

Which statement is true about \(c\vec{v}\) when \(c = 0\)?

Question 5 of 5

If \(\vec{w} = \langle 6, -2\rangle\) and \(c\vec{w} = \langle -18, 6\rangle\), what is \(c\)?