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Pre-Calculus Vectors

Vector addition (component, parallelogram, end-to-end)

20 practice questions 0 video lessons Theory + worked examples

Vector Addition

Texas Precalculus (TEKS) • Standard P.4(J) • Vectors

Vector Addition is a topic in Vectors in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.4(J), which requires students to represent and apply vector addition.

Vectors add component by component, or geometrically tip-to-tail or by the parallelogram rule, giving a single resultant.

Texas Precalculus (TEKS) › Vectors › Vector Addition  —  Standard P.4(J)

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Theory

Vectors add in three equivalent ways:

  • Component: \(\langle a_1,b_1\rangle+\langle a_2,b_2\rangle=\langle a_1+a_2,\ b_1+b_2\rangle\).
  • Tip-to-tail: place the second vector's tail at the first's tip; the sum runs start to finish.
  • Parallelogram: draw both from one point; the diagonal is the sum.

The sum is called the resultant.

All three agree. The geometry (tip-to-tail, parallelogram) is just the picture of adding components.
Tip-to-tail vector addition Placing v at the tip of u, the sum u plus v runs from the tail of u to the tip of v. u v u+v
Tip-to-tail: \(u+v\) runs from the start of \(u\) to the tip of \(v\).
Parallelogram rule Drawing u and v from the same point, the diagonal of the parallelogram they span is the sum. u v u+v
Parallelogram rule: the diagonal is \(u+v\).

Component addition and subtraction:

\[\langle a_1,b_1\rangle\pm\langle a_2,b_2\rangle=\langle a_1\pm a_2,\ b_1\pm b_2\rangle\]
add or subtract vectors by adding or subtracting corresponding components
The resultant's magnitude is \(\sqrt{a^2+b^2}\) of the sum — not the sum of the magnitudes.

How to add vectors

  1. Write both in component form.
  2. Add (or subtract) corresponding components.
  3. Find magnitude/direction of the resultant if needed.
  4. Sketch tip-to-tail to check the result makes sense.
Example 1 — Add in component form
Add \(u=\langle 3,1\rangle\) and \(v=\langle -1,4\rangle\).
Solution

Add corresponding components.

\(u+v\)\(=\)\(\langle 3+(-1),\ 1+4\rangle\)
\(=\)\(\langle 2,5\rangle\)
sum is 2 comma 5
Example 2 — Subtract vectors
Compute \(u-v\) for \(u=\langle 5,2\rangle\), \(v=\langle 1,6\rangle\).
Solution

Subtract components.

\(u-v\)\(=\)\(\langle 5-1,\ 2-6\rangle\)
\(=\)\(\langle 4,-4\rangle\)
difference is 4 comma negative 4
Example 3 — Magnitude of a resultant
Find the magnitude of \(u+v\) where \(u+v=\langle 2,5\rangle\).
Solution

Take the magnitude of the resultant.

\(\|u+v\|\)\(=\)\(\sqrt{2^2+5^2}\)
\(=\)\(\sqrt{29}\approx 5.39\)
magnitude of the resultant is root 29, about 5.39
Example 4 — Resultant from a diagram
Two forces \(\langle 6,0\rangle\) and \(\langle 0,8\rangle\) act on a point. Find the resultant's magnitude and direction.
Solution

Add, then find magnitude and direction.

\(R\)\(=\)\(\langle 6,8\rangle\)
\(\|R\|\)\(=\)\(\sqrt{36+64}=10\)
\(\theta\)\(=\)\(\arctan\dfrac{8}{6}\approx 53.1^\circ\)
resultant is 10 at about 53.1 degrees

Common pitfalls

Magnitudes don't add. \(\|u+v\|\neq\|u\|+\|v\|\) unless the vectors point the same way.
Line up components. Add \(x\) with \(x\), \(y\) with \(y\).
Subtraction reverses a vector. \(u-v=u+(-v)\).

Frequently asked questions

How do you add two vectors?

Add their corresponding components: \(\langle a_1,b_1\rangle+\langle a_2,b_2\rangle=\langle a_1+a_2,b_1+b_2\rangle\).

What is the resultant?

The single vector equal to the sum of two or more vectors — the diagonal in the parallelogram picture.

Do the magnitudes add when you add vectors?

No, unless the vectors point the same direction. Find the resultant's magnitude from its components.

What is the difference between the tip-to-tail and parallelogram methods?

None in result — both give the same sum. Tip-to-tail chains the vectors; the parallelogram draws them from a shared start.

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