The vector equation of a line Problem set A Answers

1. (a) \(\mathbf{r}=\begin{pmatrix}2\\5\end{pmatrix}+\lambda\begin{pmatrix}6\\-6\end{pmatrix}\)
2. (b) (5, 2)

1. (a) \(\mathbf{r}=\begin{pmatrix}3\\-2\\4\end{pmatrix}+\lambda\begin{pmatrix}1\\5\\-2\end{pmatrix}\)
2. (b) \(\mathbf{r}=\begin{pmatrix}4\\3\\2\end{pmatrix}+\mu\begin{pmatrix}1\\5\\-2\end{pmatrix}\)

\(\mathbf{r}=\begin{pmatrix}1\\-2\\5\end{pmatrix}+\lambda\begin{pmatrix}2\\6\\-4\end{pmatrix}\); yes, \(\lambda=\frac{3}{2}\) fits every component

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

1. (a) \(\mathbf{r}=\begin{pmatrix}0\\0\\30\end{pmatrix}+\lambda\begin{pmatrix}80\\60\\-20\end{pmatrix}\)
2. (b) Yes, \(\lambda=\frac{3}{2}\)

1. (a) \(\mathbf{r}=\begin{pmatrix}2\\-1\\3\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-2\end{pmatrix}\)
2. (b) Yes, \(\lambda=5\)

1. (a) \(\mathbf{r}=\begin{pmatrix}0\\0\\40\end{pmatrix}+\lambda\begin{pmatrix}120\\50\\-40\end{pmatrix}\)
2. (b) (90, 37.5, 10)

\(k=5\)

1. (a) \(\mathbf{r}=\begin{pmatrix}1\\-3\\2\end{pmatrix}+\lambda\begin{pmatrix}4\\6\\-8\end{pmatrix}\)
2. (b) Midpoint (3, 0, −2), at \(\lambda=\frac{1}{2}\)
3. (c) (13, 15, −22)