Scalar Multiplication Calculator

Published on: February 15, 2026
Final Answer: Free • Full Steps: Pro

This Scalar Multiplication Calculator multiplies a vector by a single number, and shows the working line by line. Every component is scaled by the same factor, which stretches or shrinks the vector without turning it. A negative scalar also flips it to point the opposite way. Following the same pattern every time makes it clear that direction only ever changes when the scalar is negative.

Step-by-step method

  1. Set up the problem.
  2. Write the scalar multiplication rule.
  3. Substitute values into the rule.
  4. Multiply each component and simplify.
See Example 1 Hide Example 1

Problem

\(k = 3,\; \vec{A} = \left\langle 1,\; 2,\; 3 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the scalar multiplication rule.
  3. Substitute values into the rule.
  4. Multiply each component and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: The scalar is \(k = 3\) and the vector is \(\vec{A} = \left\langle 1,\; 2,\; 3 \right\rangle\).

\(k = 3,\; \vec{A} = \left\langle 1,\; 2,\; 3 \right\rangle\)

Step 2

Step 2 - Write the scalar multiplication rule.

In this problem: Multiply every component of the vector by the scalar.

\(k\vec{A} = \left\langle k a_{1},\; k a_{2},\; k a_{3} \right\rangle\)

Step 3

Step 3 - Substitute values into the rule.

In this problem: Substitute the scalar and each component.

\(k\vec{A} = \left\langle 3 \cdot 1,\; 3 \cdot 2,\; 3 \cdot 3 \right\rangle\)

Step 4

Step 4 - Multiply each component and simplify.

In this problem: The resulting vector is \(\left\langle 3,\; 6,\; 9 \right\rangle\).

\(k\vec{A} = \left\langle 3,\; 6,\; 9 \right\rangle\)

Final Answer

\(k\vec{A} = \left\langle 3,\; 6,\; 9 \right\rangle\)
See Example 2 Hide Example 2

Problem

\(k = -2,\; \vec{A} = \left\langle \frac{1}{2},\; 0,\; 4 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the scalar multiplication rule.
  3. Substitute values into the rule.
  4. Multiply each component and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: The scalar is \(k = -2\) and the vector is \(\vec{A} = \left\langle \frac{1}{2},\; 0,\; 4 \right\rangle\).

\(k = -2,\; \vec{A} = \left\langle \frac{1}{2},\; 0,\; 4 \right\rangle\)

Step 2

Step 2 - Write the scalar multiplication rule.

In this problem: Multiply every component of the vector by the scalar.

\(k\vec{A} = \left\langle k a_{1},\; k a_{2},\; k a_{3} \right\rangle\)

Step 3

Step 3 - Substitute values into the rule.

In this problem: Substitute the scalar and each component.

\(k\vec{A} = \left\langle \left(-2\right) \cdot \frac{1}{2},\; \left(-2\right) \cdot 0,\; \left(-2\right) \cdot 4 \right\rangle\)

Step 4

Step 4 - Multiply each component and simplify.

In this problem: The resulting vector is \(\left\langle -1,\; 0,\; -8 \right\rangle\).

\(k\vec{A} = \left\langle -1,\; 0,\; -8 \right\rangle\)

Final Answer

\(k\vec{A} = \left\langle -1,\; 0,\; -8 \right\rangle\)

Example Problems

Click one to load it into the box and solve it.