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Section 11.2 : Vector Arithmetic
2. Given \(\vec u = 8\vec i - \vec j + 3\vec k\) and \(\vec v = 7\vec j - 4\vec k\) compute each of the following.- \( - 3\vec v\)
- \(12\vec u + \vec v\)
- \(\left\| { - 9\vec v - 2\vec u} \right\|\)
This is just a scalar multiplication problem. Just remember to multiply each component by the scalar, -3 in this case.
\[ - 3\vec v = - 3\left( {7\vec j - 4\vec k} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 21\vec j + 12\vec k}}\]b \(12\vec u + \vec v\) Show Solution
Here we’ll just do each of the scalar multiplications and then do the subtraction. With the addition just remember to add corresponding components from each vector.
\[12\vec u + \vec v = 12\left( {8\vec i - \vec j + 3\vec k} \right) + \left( {7\vec j - 4\vec k} \right) = \left( {96\vec i - 12\vec j + 36\vec k} \right) + \left( {7\vec j - 4\vec k} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{96\vec i - 5\vec j + 32\vec k}}\]c \(\left\| { - 9\vec v - 2\vec u} \right\|\) Show Solution
So, first we compute the vector inside the magnitude bars and the compute the magnitude.
\[\begin{align*} - 9\vec v - 2\vec u & = - 9\left( {7\vec j - 4\vec k} \right) - 2\left( {8\vec i - \vec j + 3\vec k} \right)\\ & = \left( { - 63\vec j + 36\vec k} \right) - \left( {16\vec i - 2\vec j + 6\vec k} \right) = - 16\vec i - 61\vec j + 30\vec k\end{align*}\]Be careful with the lack of an \(\vec i\) component in the first vector here. Just recall that means the coefficient of \(\vec i\) in the first vector is just zero!
The magnitude is then,
\[\left\| { - 9\vec v - 2\vec u} \right\| = \sqrt {{{\left( { - 16} \right)}^2} + {{\left( { - 61} \right)}^2} + {{\left( {30} \right)}^2}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\sqrt {4877} }}\]