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 $\mathbf{v}=\langle-3,6\rangle$.
\[
\begin{array}{l}
\|-4 \mathbf{v}\| \\
\|-4 \mathbf{v}\|=16
\end{array}
\]
$\square$
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Solution
Solution Steps
To find the magnitude of the scalar multiple −4v, we need to first calculate the scalar multiple of the vector v and then find the magnitude of the resulting vector.
Multiply each component of the vector v by −4.
Calculate the magnitude of the resulting vector using the formula for the magnitude of a vector x2+y2.
Step 1: Calculate the Scalar Multiple
Given the vector v=⟨−3,6⟩ and the scalar −4, we calculate the scalar multiple:
−4v=−4⟨−3,6⟩=⟨12,−24⟩
Step 2: Calculate the Magnitude
Next, we find the magnitude of the vector ⟨12,−24⟩ using the formula for the magnitude of a vector:
∥u∥=x2+y2
Substituting the components:
∥⟨12,−24⟩∥=122+(−24)2=144+576=720
Calculating 720 gives:
720=26.8328(rounded to four significant digits)
Final Answer
The magnitude of the scalar multiple −4v is
26.8328