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Pre-Calculus Vectors

Scalar multiplication of vectors

20 practice questions 0 video lessons Theory + worked examples

Scalar Multiplication of Vectors

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

Scalar Multiplication of Vectors 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 scalar multiplication of vectors.

Scalar multiplication \(k\langle a,b\rangle=\langle ka,kb\rangle\) scales a vector's length by \(|k|\) and reverses its direction when \(k<0\).

Texas Precalculus (TEKS) › Vectors › Scalar Multiplication of Vectors  —  Standard P.4(J)

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Theory

Multiplying a vector by a scalar (a number) \(k\) scales each component:

\[k\langle a,b\rangle=\langle ka,\ kb\rangle.\]

The magnitude scales by \(|k|\): \(\|kv\|=|k|\,\|v\|\). If \(k>0\) the direction is unchanged; if \(k<0\) the direction reverses.

Scalar multiplication changes length (and possibly direction), never the line of action unless \(k<0\) flips it to the opposite way.
Scalar multiplication scales a vector Multiplying a vector by 2 doubles its length in the same direction. v 2v
\(2v\) is twice as long as \(v\), same direction.
A negative scalar reverses direction Multiplying by negative one reverses the vector's direction while keeping its length. v −v
\(-v\) has the same length as \(v\) but points the opposite way.

Scalar multiplication and its effect on magnitude:

\[k\langle a,b\rangle=\langle ka,kb\rangle,\qquad \|kv\|=|k|\,\|v\|\]
k times a vector multiplies each component by k; the magnitude scales by the absolute value of k
To reach a target length \(L\): scale the unit vector, \(L\hat v\).

How to scalar-multiply

  1. Multiply each component by the scalar.
  2. Track direction: a negative scalar reverses it.
  3. Magnitude: multiply the original magnitude by \(|k|\).
  4. For a set length: scale the unit vector to that length.
Example 1 — Scale a vector
Compute \(3v\) for \(v=\langle 2,-1\rangle\).
Solution

Multiply each component by 3.

\(3v\)\(=\)\(\langle 3\cdot 2,\ 3\cdot(-1)\rangle\)
\(=\)\(\langle 6,-3\rangle\)
3 v is 6 comma negative 3
Example 2 — A negative scalar
Compute \(-2v\) for \(v=\langle 4,3\rangle\), and its magnitude.
Solution

Multiply components by \(-2\); magnitude scales by \(|-2|=2\).

\(-2v\)\(=\)\(\langle -8,-6\rangle\)
\(\|-2v\|\)\(=\)\(\sqrt{64+36}=10\)

Direction reverses; magnitude is \(2\|v\|=2(5)=10\).

negative 2 v is negative 8 comma negative 6, magnitude 10
Example 3 — Combine operations
For \(u=\langle 1,2\rangle\), \(v=\langle 3,-1\rangle\), find \(2u+v\).
Solution

Scale, then add.

\(2u\)\(=\)\(\langle 2,4\rangle\)
\(2u+v\)\(=\)\(\langle 2+3,\ 4-1\rangle\)
\(=\)\(\langle 5,3\rangle\)
2 u plus v is 5 comma 3
Example 4 — Scale to a required length
Find a vector of magnitude \(15\) in the direction of \(v=\langle 3,4\rangle\).
Solution

Take the unit vector, then multiply by 15.

\(\hat v\)\(=\)\(\dfrac{1}{5}\langle 3,4\rangle\)
\(15\hat v\)\(=\)\(\langle 9,12\rangle\)
the vector is 9 comma 12

Common pitfalls

Multiply every component. Don't scale just one.
A negative scalar flips direction. The magnitude still uses \(|k|\).
Magnitude scales by \(|k|\), not \(k^2\) — \(\|3v\|=3\|v\|\).

Frequently asked questions

What is scalar multiplication of a vector?

Multiplying each component by a number \(k\): \(k\langle a,b\rangle=\langle ka,kb\rangle\).

What does a negative scalar do?

It reverses the vector's direction while scaling its length by the absolute value of the scalar.

How does scaling affect magnitude?

\(\|kv\|=|k|\,\|v\|\): the magnitude multiplies by the absolute value of the scalar.

How do you make a vector a specific length?

Find the unit vector and multiply by the desired length \(L\): \(L\hat v\).

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