Chapter 1: Vectors, Lines and Planes

Section 1.3: Dot Product

Example 1.3.5

$$ If $\mathbf{A}\=3 \mathbf{i}\+2 \mathbf{j}\+5 \mathbf{k}$ and $\mathbf{B}\=-4 \mathbf{i}\+5 \mathbf{j}\+\mathrm{\λ} \mathbf{k}$, find all values of  $\mathrm{\λ}$ that make B orthogonal to A. $$ Solution Mathematical Solution $$ For B to be orthogonal to A, the following must hold.   $\mathbf{A}·\mathbf{B}\=\left[\begin{array}{c}3\\ 2\\ 5\end{array}\right]·\left[\begin{array}{c}-4\\ 5\\ \mathrm{\λ}\end{array}\right]\=-12\+10\+5 \mathrm{\λ}$ = $5 \mathrm{\λ}-2\=0$ $$ Consequently, $\mathrm{\λ}\=2\/5$. $$$$ Maple Solution $$ Enter the vectors A and B •  Enter A as per Table 1.1.1.a •  Context Panel: Assign Name $\mathbf{A}\=⟨3\,2\,5⟩$$\stackrel{\text{assign}}{\to }$ •  Enter B as per Table 1.1.1. •  Context Panel: Assign Name $\mathbf{B}\=⟨-4\,5\,\mathrm{\λ}⟩$$\stackrel{\text{assign}}{\to }$ Form and solve the equation $\mathbf{A}·\mathbf{B}\=0$  •  Common Symbols palette: Dot-product operator Press the Enter key.   •  Context Panel: Solve≻Solve $\mathbf{A}·\mathbf{B}\=0$ $5\mathrm{\λ}-2\=0$ $\stackrel{\text{solve}}{\to }$ $\left\{\mathrm{\λ}\=\frac{2}{5}\right\}$ $$

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