Calculators / Calculus 3 / Scalar Multiplication

Scalar Multiplication

Published on: February 15, 2026

This Scalar Multiplication Calculator helps you multiply a 3D vector by a scalar. Multiply each component of the vector by the same scalar value, then write the results together as a new vector. This follows the scalar multiplication rule used in coordinate geometry and vector algebra. It is a simple way to check answers, understand the method clearly, and practise vector operations step by step.

Step-by-step method

  1. Identify the scalar and the vector components.
  2. Write the scalar multiplication rule.
  3. Substitute values into the rule.
  4. Multiply each component and simplify.

Formula:

k·A=( k·a1, k·a2, k·a3 )

Example 1: 3,(1,2,3)

Step 1 - Identify the values.

In this problem: Write the scalar and vector components.

k=3
a1=1, a2=2, a3=3

Step 2 - Write the rule.

In this problem: Multiply the scalar with each component.

k·A=( k·a1, k·a2, k·a3 )

Step 3 - Substitute values.

In this problem: Replace k and the components with your values.

3·A=( 3·1, 3·2, 3·3 )

Step 4 - Multiply and simplify.

In this problem: Compute each product.

3·A=( 3, 6, 9 )

Final answer: k·A = ( 3, 6, 9 )

Example 2: -2,(1/2,0,4)

Step 1 - Identify the values.

In this problem: Write the scalar and vector components.

k=-2
a1=
1
2
, a2=0, a3=4

Step 2 - Write the rule.

In this problem: Multiply the scalar with each component.

k·A=( k·a1, k·a2, k·a3 )

Step 3 - Substitute values.

In this problem: Replace k and the components with your values.

-2·A=( -2·
1
2
, -2·0, -2·4 )

Step 4 - Multiply and simplify.

In this problem: Compute each product.

-2·A=( -1, 0, -8 )

Final answer: k·A = ( -1, 0, -8 )