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Vector Subtraction
Graphically

Learn to subtract vectors using the additive inverse, represent the result by connecting tips, and compute component-wise differences.

Precalculus

What Is a Vector?

A vector is a quantity with both magnitude (length) and direction. We write vectors as ordered pairs of components: \(\mathbf{v} = \langle v_1, v_2 \rangle\).

\(x\)
\(y\)
\(\mathbf{v}\)
\(v_1\) (horizontal)
\(v_2\)
(vertical)
\(O\)

The Additive Inverse \(-\mathbf{w}\)

For any vector \(\mathbf{w} = \langle w_1, w_2 \rangle\), its additive inverse is \(-\mathbf{w} = \langle -w_1, -w_2 \rangle\). It has the same magnitude but points in the opposite direction.

\(\mathbf{w}\)
\(-\mathbf{w}\)
\(O\)
\(|\mathbf{w}|\)
\(|\mathbf{w}|\)

What Is Vector Subtraction?

Vector subtraction is defined in terms of vector addition:

This means: to subtract \(\mathbf{w}\) from \(\mathbf{v}\), reverse \(\mathbf{w}\) and then add it to \(\mathbf{v}\) using the parallelogram or tip-to-tail method.

Step 1
Negate \(\mathbf{w}\): flip its direction
Step 2
Add \(\mathbf{v} + (-\mathbf{w})\) tip-to-tail
Result
The resultant is \(\mathbf{v} - \mathbf{w}\)

Component-Wise Subtraction

Subtract matching components: horizontal from horizontal, vertical from vertical. This works because negating \(\mathbf{w}\) negates each component, then we add.

Horizontal
\(v_1 - w_1\)
Vertical
\(v_2 - w_2\)

Graphical: Tip-to-Tip Method

When \(\mathbf{v}\) and \(\mathbf{w}\) share the same initial point, the vector \(\mathbf{v} - \mathbf{w}\) points from the tip of \(\mathbf{w}\) to the tip of \(\mathbf{v}\).

\(\mathbf{v}\)
\(\mathbf{w}\)
\(\mathbf{v}-\mathbf{w}\)
\(O\)
tip of v
tip of w

Graphical: Adding the Inverse

Alternatively, translate \(-\mathbf{w}\) so its tail meets \(\mathbf{v}\)'s tip, then draw from the origin to the new endpoint. This is the standard tip-to-tail addition of \(\mathbf{v} + (-\mathbf{w})\).

\(\mathbf{v}\)
\(-\mathbf{w}\)
\(\mathbf{v}-\mathbf{w}\)
\(O\)

Addition vs Subtraction

\(\mathbf{v} + \mathbf{w}\) \(\mathbf{v} - \mathbf{w}\)
Definition \(\langle v_1+w_1,\;v_2+w_2\rangle\) \(\langle v_1-w_1,\;v_2-w_2\rangle\)
Graphical Tip-to-tail: tail of w at tip of v Tip-to-tip: from tip of w to tip of v
Think of it as Combining motions Finding the difference between positions
Direction of result Along the diagonal of the parallelogram Along the other diagonal

How to Subtract Vectors

1
Write components. Identify \(\mathbf{v}=\langle v_1,v_2\rangle\) and \(\mathbf{w}=\langle w_1,w_2\rangle\).
2
Subtract component-wise. Compute \(\langle v_1-w_1,\;v_2-w_2\rangle\).
3
Draw (optional). Place both vectors at the origin. Draw from the tip of \(\mathbf{w}\) to the tip of \(\mathbf{v}\).
4
Verify. Check that \(\mathbf{w}+(\mathbf{v}-\mathbf{w})=\mathbf{v}\).

Example 1: Simple Subtraction

Let \(\mathbf{v}=\langle 5,3\rangle\) and \(\mathbf{w}=\langle 2,1\rangle\). Find \(\mathbf{v}-\mathbf{w}\).

1
2
3
4
5
1
2
3
\(\mathbf{v}\)
\(\mathbf{w}\)
\(\mathbf{v}-\mathbf{w}\)

Example 2: Negative Components

Let \(\mathbf{v}=\langle 1, 4\rangle\) and \(\mathbf{w}=\langle 3, -1\rangle\). Find \(\mathbf{v}-\mathbf{w}\).

\(\mathbf{v}=\langle 1,4\rangle\)
\(\mathbf{w}=\langle 3,-1\rangle\)
\(\mathbf{v}-\mathbf{w}\)
\(O\)

Example 3: Verify with Addition

Given \(\mathbf{a}=\langle -3,7\rangle\) and \(\mathbf{b}=\langle 4,2\rangle\), find \(\mathbf{a}-\mathbf{b}\) and verify.

1
Subtract: \(\mathbf{a}-\mathbf{b}=\langle -3-4,\;7-2\rangle = \langle -7,\;5\rangle\)
2
Verify: \(\mathbf{b}+(\mathbf{a}-\mathbf{b})=\langle 4,2\rangle+\langle -7,5\rangle = \langle -3,7\rangle = \mathbf{a}\) ✓

The Parallelogram View

When you place \(\mathbf{v}\) and \(\mathbf{w}\) at the same origin, they form two sides of a parallelogram. The two diagonals are \(\mathbf{v}+\mathbf{w}\) and \(\mathbf{v}-\mathbf{w}\).

\(\mathbf{v}\)
\(\mathbf{w}\)
\(\mathbf{v}+\mathbf{w}\)
\(\mathbf{v}-\mathbf{w}\)
\(O\)

Why the Tip-to-Tip Rule Works

1
Start with the definition: \(\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})\).
2
Rearrange: \(\mathbf{w}+(\mathbf{v}-\mathbf{w})=\mathbf{v}\). This means the difference vector, placed at \(\mathbf{w}\)'s tip, reaches \(\mathbf{v}\)'s tip.
3
Therefore: The vector from \(\mathbf{w}\)'s tip to \(\mathbf{v}\)'s tip is \(\mathbf{v}-\mathbf{w}\).

Common Mistakes to Avoid

✕ Wrong arrow direction
Drawing from \(\mathbf{v}\)'s tip to \(\mathbf{w}\)'s tip gives \(\mathbf{w}-\mathbf{v}\), not \(\mathbf{v}-\mathbf{w}\). Order matters.
✕ Subtracting magnitudes only
You cannot compute \(|\mathbf{v}|-|\mathbf{w}|\) and call it the difference. Subtract each component separately.
✕ Sign errors with negatives
Forgetting that \(a - (-b) = a+b\). When a component is negative, subtraction becomes addition. E.g., \(5 - (-4) = 9\).

Summary of Methods

Method Algebraic Graphical (Tip-to-Tip) Graphical (Inverse)
Process \(\langle v_1-w_1,\;v_2-w_2\rangle\) Arrow from tip of w to tip of v Flip w, then tip-to-tail add
Best for Exact computation Quick sketching Understanding the concept
Verify \(\mathbf{w}+\text{result}=\mathbf{v}\) Arrow endpoints match Matches algebraic answer

Question 1 of 5

If \(\mathbf{v}=\langle 6,2\rangle\) and \(\mathbf{w}=\langle 4,5\rangle\), what is \(\mathbf{v}-\mathbf{w}\)?

Question 2 of 5

The additive inverse \(-\mathbf{w}\) of a vector \(\mathbf{w}\) has:

Question 3 of 5

To graphically represent \(\mathbf{v}-\mathbf{w}\) using the tip-to-tip method, you draw an arrow from:

Question 4 of 5

If \(\mathbf{a}=\langle -2,5\rangle\) and \(\mathbf{b}=\langle 3,-4\rangle\), what is \(\mathbf{a}-\mathbf{b}\)?

Question 5 of 5

Which equation correctly expresses the relationship that defines vector subtraction?