Problem 58 - College Algebra

Question

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

Step-by-Step Solution

Answer

Scalar multiplication requires each component of the vector to be individually multiplied by the scalar. For instance, the scalar multiplication of the scalar 2 and the vector \(\langle 3, 4 \rangle\) will give a result vector \(\langle 6, 8 \rangle\).

Step 1: Understanding Scalar Multiplication

From the basics of linear algebra, we know that scalar multiplication is the product of a scalar (a real number) and a vector. This can be represented as \(r \cdot \mathbf{v}\) where \(r\) is a real number (scalar) and \(\mathbf{v}\) is a vector.

Step 2: Performing Scalar Multiplication

Before performing a scalar multiplication, ensure you know the components of your vector as well as the scalar. If we have a vector \(\mathbf{v} = \langle a, b \rangle\) and a scalar \(r\), the scalar multiplication is computed by multiplying each component of the vector by the scalar. In other words, \(r \cdot \mathbf{v} = \langle r \cdot a, r \cdot b \rangle\). Set \(r = 2\) and \(\mathbf{v} = \langle 3, 4 \rangle\) as an example.

Step 3: Calculating the Scalar Multiplication

Now, execute the multiplication we have set up in the previous step. In this instance, our work is \(2 \cdot \langle 3, 4 \rangle = \langle 2 \cdot 3, 2 \cdot 4 \rangle\). After performing the multiplication, we find that \(2 \cdot \langle 3, 4 \rangle = \langle 6, 8 \rangle\).