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Add Vectors in
Different Ways

Master end-to-end, component-wise, and parallelogram methods for vector addition — and discover why \(\lVert\mathbf{u}+\mathbf{v}\rVert \neq \lVert\mathbf{u}\rVert+\lVert\mathbf{v}\rVert\) in general.

Precalculus

What Is a Vector?

A vector is a quantity that has both magnitude (length) and direction. We write vectors as \(\mathbf{v}\) or \(\vec{v}\), and represent them with arrows.

\(\mathbf{u}\)
\(\mathbf{v}\)
magnitude + direction
magnitude + direction

Why Add Vectors?

In physics and engineering, forces, velocities, and displacements combine as vectors. The resultant vector \(\mathbf{u}+\mathbf{v}\) tells you the net effect of two actions applied together.

Physics
Two forces on an object produce a single net force: their vector sum.
Navigation
A plane's velocity plus wind velocity gives the actual ground velocity.
Gaming
Character movement is the vector sum of directional inputs.

Three Methods of Vector Addition

1. End-to-End
Place the tail of \(\mathbf{v}\) at the tip of \(\mathbf{u}\). The resultant runs from the tail of \(\mathbf{u}\) to the tip of \(\mathbf{v}\).
2. Component-Wise
Add matching components: \(\langle u_1+v_1,\; u_2+v_2 \rangle\). Purely algebraic.
3. Parallelogram
Place both tails together. Complete the parallelogram. The diagonal is the sum.

End-to-End: Tip-to-Tail

\(\mathbf{u}\)
\(\mathbf{v}\)
\(\mathbf{u}+\mathbf{v}\)
tail of u
tip of v

Component-Wise Addition

Given \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\):

x
y
\(\mathbf{u} = \langle 3, 1 \rangle\)
\(\mathbf{v} = \langle 1, 2 \rangle\)
\(\mathbf{u+v} = \langle 4, 3 \rangle\)
1
2
3
4
1
2
3

The Parallelogram Rule

\(\mathbf{u}\)
\(\mathbf{v}\)
\(\mathbf{u}+\mathbf{v}\)
common tail

Magnitude: Not Simply Additive

Think of the triangle inequality: the sum of two sides of a triangle is always greater than the third side. The resultant is the "third side," so it must be shorter than the other two combined (unless they are collinear).

\(\lVert\mathbf{u}\rVert = 5\)
\(\lVert\mathbf{v}\rVert = 4\)
\(\lVert\mathbf{u}+\mathbf{v}\rVert \approx 6.4 < 9\)

Scalar vs Vector Addition

Scalar Addition Vector Addition
What you add Single numbers Ordered pairs / arrows
Direction Not applicable Must be considered
Result magnitude \(|a+b| = |a|+|b|\) when signs agree \(\lVert\mathbf{u}+\mathbf{v}\rVert \leq \lVert\mathbf{u}\rVert+\lVert\mathbf{v}\rVert\)
Visualization Number line Coordinate plane / space

Properties of Vector Addition

1
Commutative: \(\mathbf{u}+\mathbf{v} = \mathbf{v}+\mathbf{u}\)
2
Associative: \((\mathbf{u}+\mathbf{v})+\mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w})\)
3
Zero vector: \(\mathbf{u}+\mathbf{0} = \mathbf{u}\)
4
Additive inverse: \(\mathbf{u}+(-\mathbf{u}) = \mathbf{0}\)

How to Add Vectors: Step by Step

1
Identify components. Write each vector as \(\langle x, y \rangle\) or read them from the diagram.
2
Add component-wise. Sum the \(x\)-components and sum the \(y\)-components separately.
3
Find the magnitude. Use \(\lVert\mathbf{u}+\mathbf{v}\rVert = \sqrt{(u_1+v_1)^2 + (u_2+v_2)^2}\).
4
Find the direction. Use \(\theta = \tan^{-1}\!\left(\frac{u_2+v_2}{u_1+v_1}\right)\) and adjust for the correct quadrant.
5
Verify graphically. Sketch end-to-end or parallelogram to check reasonableness.

Example 1: Component-Wise

Find \(\mathbf{u}+\mathbf{v}\) given \(\mathbf{u} = \langle 3, 2 \rangle\) and \(\mathbf{v} = \langle -1, 4 \rangle\).

1
Add \(x\)-components: \(3 + (-1) = 2\)
2
Add \(y\)-components: \(2 + 4 = 6\)
3
Result: \(\mathbf{u}+\mathbf{v} = \langle 2, 6 \rangle\)

Example 2: End-to-End Graphically

Add \(\mathbf{u} = \langle 4, 1 \rangle\) and \(\mathbf{v} = \langle -2, 3 \rangle\) using tip-to-tail.

x
y
\(\mathbf{u} = \langle 4, 1 \rangle\)
\(\mathbf{v} = \langle -2, 3 \rangle\)
\(\mathbf{u+v} = \langle 2, 4 \rangle\)
1
2
3
4
1
2
3
4

Example 3: Parallelogram with Magnitude

Let \(\mathbf{u} = \langle 5, 0 \rangle\) and \(\mathbf{v} = \langle 2, 4 \rangle\). Find \(\mathbf{u}+\mathbf{v}\), its magnitude, and its direction.

\(\mathbf{u} = \langle 5, 0 \rangle\)
\(\mathbf{v} = \langle 2, 4 \rangle\)
\(\langle 7, 4 \rangle\)

Common Mistakes

Adding Magnitudes
Assuming \(\lVert\mathbf{u}+\mathbf{v}\rVert = \lVert\mathbf{u}\rVert+\lVert\mathbf{v}\rVert\). This ignores direction and only works when vectors are parallel and same direction.
Wrong Placement
In tip-to-tail, placing both tails at the origin instead of chaining the second vector from the first's tip.
Sign Errors
Forgetting that leftward or downward components are negative. Always keep track of signs in \(\langle x, y \rangle\).

Summary: Three Methods Compared

Method When to Use Key Step
End-to-End Quick geometric sketch Chain tail to tip
Component-Wise Precise calculation Add \(x\) and \(y\) separately
Parallelogram Both vectors share a point Draw diagonal from common tail

Question 1 of 5

If \(\mathbf{u} = \langle 3, -2 \rangle\) and \(\mathbf{v} = \langle -1, 5 \rangle\), what is \(\mathbf{u}+\mathbf{v}\)?

Question 2 of 5

In the tip-to-tail method, where do you place the tail of the second vector?

Question 3 of 5

If \(\lVert\mathbf{u}\rVert = 5\) and \(\lVert\mathbf{v}\rVert = 12\) and the two vectors point in the same direction, what is \(\lVert\mathbf{u}+\mathbf{v}\rVert\)?

Question 4 of 5

What is the magnitude of \(\langle 3, 4 \rangle + \langle 1, -1 \rangle\)?

Question 5 of 5

In the parallelogram rule, the resultant \(\mathbf{u}+\mathbf{v}\) is represented by which part of the parallelogram?