Introduction

• Scalar multiplication is the process of taking a vector and multiplying it by a number (a scalar).
• This operation changes the size (length) of the vector and may also change its direction.

What It Means to “Scale” a Vector

• A scalar is just a real number like $2$, $-3$, or $0.5$.
• To scale a vector means to multiply every component of the vector by that scalar.
• If $ \vec{v} = (v_1, v_2) $, then $$k\vec{v} = (kv_1, kv_2).$$
• Think of it like adjusting the volume on a speaker: the shape of the sound stays the same, but its intensity changes.

Positive Scaling: Stretching and Shrinking

• When $k > 0$, the vector keeps its direction.
• The vector becomes:
  • Longer if $k > 1$
  • Shorter if $0 < k < 1$
• Example:
  • $2(3,1) = (6,2)$ stretches the vector to twice its length.
  • $0.5(3,1) = (1.5,0.5)$ shrinks it to half its length.

Negative Scaling: Reversing Direction

• When $k < 0$, the vector flips to point in the opposite direction.
• The size still changes according to $|k|$.
• Example:
  • $-1(3,1) = (-3,-1)$ flips the vector but keeps the same length.
  • $-2(3,1) = (-6,-2)$ flips and stretches it.

Zero as a Special Case

• Multiplying any vector by $0$ gives the zero vector: $$0\vec{v} = (0,0).$$
• The zero vector has no direction and no length.
• This is the only time scalar multiplication destroys direction entirely.

Real‑World Examples of Scaling

• Physics: Doubling a force vector doubles the push in the same direction.
• Graphics: Shrinking an object uniformly is like scaling all its position vectors by $0.5$.
• Navigation: Reversing direction is like turning around and walking the same path backward.
• Economics: A vector representing quantities of goods can be scaled to represent bundles of different sizes.

Unit Vectors

Unit vectors are the “building blocks” of vector notation. They give a clean, simple way to describe direction without worrying about length. Once direction is fixed, any vector of any size can be built by scaling a unit vector.

What a Unit Vector Is

A unit vector is a vector whose length is exactly 1 unit.
Its only job is to show direction.

• Length (magnitude): $$|\hat{u}| = 1$$
• Direction: whatever direction the vector points
• Notation: a small “hat” over a letter, like \(\hat{u}\)

Examples:

• A unit vector pointing east
• A unit vector pointing straight up
• A unit vector pointing $45^\circ$ northeast

These all have different directions but the same length: 1 unit.

Standard Unit Vectors in the Plane

In 2‑dimensional space, two special unit vectors are used constantly:

• $\hat{i}$: one unit in the $x$-direction $$\hat{i} = (1, 0)$$
• $\hat{j}$: one unit in the $y$-direction $$\hat{j} = (0, 1)$$

Any vector in the plane can be written using these two.

Example:
A vector $(3, -2)$ can be written as: $$3\hat{i} - 2\hat{j}$$ This simply means “3 units right, 2 units down.”

Building Vectors from Unit Vectors

If you know a unit vector $\hat{u}$ that points in the direction you want, then any vector of length $L$ in that direction is: $$\vec{v} = L \hat{u}$$ Example:
A 5‑unit vector pointing in the direction of $\hat{u} = \left(\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}\right)$ is: $$5\hat{u} = \left( \frac{5\sqrt{3}}{2},\ \frac{5}{2} \right)$$

Creating a Unit Vector from Any Vector

If you have a vector $\vec{v} = (x, y)$, you can turn it into a unit vector by dividing by its length.

1. Compute the length: $$|\vec{v}| = \sqrt{x^2 + y^2}$$
2. Divide each component by the length: $$\hat{v} = \frac{1}{|\vec{v}|}(x, y)$$

This gives a vector pointing in the same direction as $\vec{v}$, but with length 1.

Example:

$$\vec{v} = (3, 4)$$ Calculate the length: $$\sqrt{3^2 + 4^2} = 5$$ Unit vector: $$\hat{v} = \left(\frac{3}{5},\ \frac{4}{5}\right)$$

Unit Vectors from Angle + Length Notation

If a vector has length $L$ and angle $\theta$, then the unit vector in that direction is: $$\hat{u} = (\cos\theta,\ \sin\theta)$$ A vector of length $L$ in that direction is: $$\vec{v} = L(\cos\theta,\ \sin\theta)$$ Example:
A 1‑unit vector at $30^\circ$: $$(\cos 30^\circ,\ \sin 30^\circ) = \left(\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$$ A 5‑unit vector at the same angle: $$5\left(\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$$

Why Unit Vectors Matter

• They make vector expressions shorter and clearer.
• They separate direction from length, which is useful in physics and geometry.
• They make it easy to scale vectors up or down.
• They help convert between component form and angle‑plus‑length form.

A natural next step is to look at how unit vectors help with vector projections and decomposing vectors into directional components—would you like a section on that?

Common Mistakes and How to Avoid Them

• Mistake: Thinking negative scalars change the angle by something other than $180^\circ$.
  • Fix: Remember: negative = flip.
• Mistake: Forgetting to multiply every component.
  • Fix: Apply the scalar to each coordinate.
• Mistake: Confusing shrinking with “losing direction.”
  • Fix: Only the zero vector loses direction.

Summary

• Scalar multiplication changes a vector’s length and possibly its direction.
• Positive scalars stretch or shrink.
• Negative scalars flip and stretch/shrink.
• Zero collapses everything to the zero vector.
• The operation is simple: multiply each component by the scalar.

Exercises

1. Compute $2(3, -1)$.

   Solution

   Compute $2(3, -1)$.

   • Multiply each component by $2$: $$2(3, -1) = (2\cdot 3,\; 2\cdot (-1)) = (6, -2).$$
2. Compute $-3(1, 4)$.

   Solution

   Compute $-3(1, 4)$.

   • Multiply each component by $-3$: $$-3(1, 4) = (-3\cdot 1,\; -3\cdot 4) = (-3, -12).$$
3. Compute $0(7, -2)$.

   Solution

   Compute $0(7, -2)$.

   • Any vector times $0$ is the zero vector: $$0(7, -2) = (0, 0).$$
4. A vector $\vec{v}$ has length $5$. What is the length of $4\vec{v}$?

   Solution

   A vector $\vec{v}$ has length $5$. What is the length of $4\vec{v}$?

   • Length scales by $|k|$: $$|4\vec{v}| = |4|\cdot|\vec{v}| = 4\cdot 5 = 20.$$
5. A vector $\vec{w}$ has length $10$. What is the length of $-0.5\vec{w}$?

   Solution

   A vector $\vec{w}$ has length $10$. What is the length of $-0.5\vec{w}$?

   • Length scales by $|k|$: $$|-0.5\vec{w}| = |-0.5|\cdot|\vec{w}| = 0.5\cdot 10 = 5.$$
6. Does the vector change direction when multiplied by $0.2$? Explain briefly.

   Solution

   Does the vector change direction when multiplied by $0.2$? Explain briefly.

   • No.
   • Since $0.2 > 0$, the direction stays the same; only the length shrinks to $0.2$ times the original.
7. Does the vector change direction when multiplied by $-1$? Explain briefly.

   Solution

   Does the vector change direction when multiplied by $-1$? Explain briefly.

   • Yes.
   • Multiplying by $-1$ flips the vector to point in exactly the opposite direction, with the same length.
8. If $\vec{v} = (2,3)$, find a scalar $k$ such that $k\vec{v} = (4,6)$.

   Solution

   If $\vec{v} = (2,3)$, find a scalar $k$ such that $k\vec{v} = (4,6)$.

   • We want: $$k(2,3) = (4,6).$$
   • Compare components: $$2k = 4 \Rightarrow k = 2,\quad 3k = 6 \Rightarrow k = 2.$$
   • So $k = 2$.
9. Describe in words what happens to a vector when multiplied by a scalar between $0$ and $1$.

   Solution

   Describe in words what happens to a vector when multiplied by a scalar between $0$ and $1$.

   • The vector keeps the same direction.
   • Its length decreases; it is shrunk toward the origin.
   • Example: multiplying by $0.5$ halves the length.
10. Describe in words what happens when a vector is multiplied by a negative scalar.

    Solution

    Describe in words what happens when a vector is multiplied by a negative scalar.

    • The vector’s direction reverses (it points the opposite way).
    • Its length is scaled by the absolute value of the scalar.
    • Example: multiplying by $-3$ flips the direction and makes the vector three times as long.