# Convert $\frac{x-5}2=y+1=\frac{z+3}6$ to vector form

Convert \frac{x-5}2=y+1=\frac{z+3}6 to vector form

• \lambda=\frac{x-5}2 \Rightarrow x=5+2\lambda
• \lambda=y+1 \Rightarrow y=-1+\lambda
• \lambda=\frac{z+3}6 \Rightarrow z=-3+6\lambda
• Answer: \vec r = \begin{pmatrix}5\\-1\\-3\end{pmatrix} + \lambda \begin{pmatrix}2\\1\\6\end{pmatrix}

You can see that the first vector corresponds to the negatives of the constant added to each position (e.g. x-5 becomes 5 in the vector), and the second vector corresponds to the denominators of each fraction (e.g. \frac{x-5}2 becomes 2 in the vector).

Convert \frac{x+2}4=\frac{y-3}5=z+1 to vector form

• \lambda=\frac{x+2}4 \Rightarrow x=-2+4\lambda
• \lambda=\frac{y-3}5 \Rightarrow y=3+5\lambda
• \lambda=z+1 \Rightarrow z=-1+\lambda
• Answer: \vec r = \begin{pmatrix}-2\\3\\-1\end{pmatrix} + \lambda \begin{pmatrix}4\\5\\1\end{pmatrix}