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

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
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Solution

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Solution Steps

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

  1. Multiply each component of the vector v\mathbf{v} by 4-4.
  2. Calculate the magnitude of the resulting vector using the formula for the magnitude of a vector x2+y2\sqrt{x^2 + y^2}.
Step 1: Calculate the Scalar Multiple

Given the vector v=3,6\mathbf{v} = \langle -3, 6 \rangle and the scalar 4-4, we calculate the scalar multiple: 4v=43,6=12,24 -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 12,24\langle 12, -24 \rangle using the formula for the magnitude of a vector: u=x2+y2 \|\mathbf{u}\| = \sqrt{x^2 + y^2} Substituting the components: 12,24=122+(24)2=144+576=720 \|\langle 12, -24 \rangle\| = \sqrt{12^2 + (-24)^2} = \sqrt{144 + 576} = \sqrt{720} Calculating 720\sqrt{720} gives: 720=26.8328(rounded to four significant digits) \sqrt{720} = 26.8328 \quad (\text{rounded to four significant digits})

Final Answer

The magnitude of the scalar multiple 4v-4 \mathbf{v} is 26.8328 \boxed{26.8328}

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