Questions: Find the magnitude of the scalar multiple, where (mathbfu=langle 2,0rangle) and (mathbfv=langle-3,6rangle). [ -4 mathbfv -4 mathbfv=16 ]

Transcript text: Find the magnitude of the scalar multiple, where $\mathbf{u}=\langle 2,0\rangle$ and $\mathbf{v}=\langle-3,6\rangle$. \[ \begin{array}{l} \|-4 \mathbf{v}\| \\ \|-4 \mathbf{v}\|=16 \end{array} \] $\square$ Need Help? Read It Submit Answer

Solution

Solution Steps

To find the magnitude of the scalar multiple $-4 \mathbf{v}$, we need to first calculate the scalar multiple of the vector $\mathbf{v}$ and then find the magnitude of the resulting vector.

Multiply each component of the vector $\mathbf{v}$ by $-4$.
Calculate the magnitude of the resulting vector using the formula for the magnitude of a vector $\sqrt{x^2 + y^2}$.

Step 1: Calculate the Scalar Multiple

Given the vector $\mathbf{v} = \langle -3, 6 \rangle$ and the scalar $-4$, we calculate the scalar multiple: \[ -4 \mathbf{v} = -4 \langle -3, 6 \rangle = \langle 12, -24 \rangle \]

Step 2: Calculate the Magnitude

Next, we find the magnitude of the vector $\langle 12, -24 \rangle$ using the formula for the magnitude of a vector: \[ \|\mathbf{u}\| = \sqrt{x^2 + y^2} \] Substituting the components: \[ \|\langle 12, -24 \rangle\| = \sqrt{12^2 + (-24)^2} = \sqrt{144 + 576} = \sqrt{720} \] Calculating $\sqrt{720}$ gives: \[ \sqrt{720} = 26.8328 \quad (\text{rounded to four significant digits}) \]