The vector equation of a line Problem set B Answers

1. (a) \(\mathbf{r}=\begin{pmatrix}-3\\4\end{pmatrix}+\lambda\begin{pmatrix}8\\-6\end{pmatrix}\)
2. (b) (1, 1)

1. (a) \(\mathbf{r}=\begin{pmatrix}-1\\6\\0\end{pmatrix}+\lambda\begin{pmatrix}4\\-3\\2\end{pmatrix}\)
2. (b) \(\mathbf{r}=\begin{pmatrix}3\\3\\2\end{pmatrix}+\mu\begin{pmatrix}4\\-3\\2\end{pmatrix}\)

\(\mathbf{r}=\begin{pmatrix}2\\3\\-4\end{pmatrix}+\lambda\begin{pmatrix}2\\-4\\2\end{pmatrix}\); yes, \(\lambda=3\) fits every component

No: \(\lambda=3\) from the first equation does not satisfy the others

1. (a) \(\mathbf{r}=\begin{pmatrix}-20\\10\\45\end{pmatrix}+\lambda\begin{pmatrix}60\\-30\\-30\end{pmatrix}\)
2. (b) No: \(\lambda=2\) fits two equations but not the third

1. (a) \(\mathbf{r}=\begin{pmatrix}-3\\4\\1\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\3\end{pmatrix}\)
2. (b) No: \(\lambda=4\) fits two equations but not the third

1. (a) \(\mathbf{r}=\begin{pmatrix}0\\0\\50\end{pmatrix}+\lambda\begin{pmatrix}80\\-60\\-50\end{pmatrix}\)
2. (b) (48, −36, 20)

\(k=11\)

1. (a) \(\mathbf{r}=\begin{pmatrix}-2\\4\\6\end{pmatrix}+\lambda\begin{pmatrix}4\\-6\\-8\end{pmatrix}\)
2. (b) Midpoint (0, 1, 2), at \(\lambda=\frac{1}{2}\)
3. (c) (10, −14, −18)