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Pre-Calculus Vectors

Scalar multiplication of vectors

20 practice questions 0 video lessons Theory + worked examples

Scalar Multiplication of Vectors

Common Core Pre-Calculus • Standard N-VM.5 • Vectors

Scalar Multiplication of Vectors is a topic in Vectors in the Common Core State Standards. It is aligned to Standard N-VM.5, which requires students to multiply a vector by a scalar and describe the effect.

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

Common Core Pre-Calculus › Vectors › Scalar Multiplication of Vectors  —  Standard N-VM.5

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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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