Problem 71 - College Algebra

Question

Describe how to perform scalar multiplication. Provide an example with your description.

Step-by-Step Solution

Answer

In scalar multiplication, each component of a vector is multiplied by a given scalar. For example, given the vector (2,3) and scalar 3, the scalar multiplication would result in the vector (6,9).

Step 1: Definition of Scalar Multiplication

Scalar multiplication is an operation that takes a scalar (a single number) and a vector, and multiplies each component of the vector by the scalar. For any vector \( \vec{v} = v_1, v_2, ..., v_n \) and scalar \( c \), scalar multiplication is defined as \( c \cdot \vec{v} = c \cdot v_1, c \cdot v_2, ..., c \cdot v_n \).

Step 2: Example of a Vector

For this explanation, let's take an example of a vector \( \vec{v} \) = (2,3). We'll also take a scalar \( c \) = 3.

Step 3: Perform Scalar Multiplication

Next, multiply each component of the vector by the scalar: \( c \cdot \vec{v} \) = 3 \cdot (2,3) = (3*2, 3*3) = (6,9).

Step 4: Verification

The final result of the scalar multiplication is the vector (6,9). Whenever you perform scalar multiplication, you should always get a vector of the same size as the original one but with its magnitude scaled by the factor of the scalar.