Determine whether the statement is true or false. Explain...

Determine whether the statement is true or false. Explain your answer. In these exercises $L_{0}$ and $L_{1}$ are lines in 3 -space whose parametric equations are $$ \begin{array}{ll}{L_{0}: x=x_{0}+a_{0} t,} & {y=y_{0}+b_{0} t, \quad z=z_{0}+c_{0} t} \\ {L_{1}: x=x_{1}+a_{1} t,} & {y=y_{1}+b_{1} t, \quad z=z_{1}+c_{1} t}\end{array} $$ If $L_{1}$ and $L_{2}$ are parallel, then $\mathbf{v}_{0}=\left\langle a_{0}, b_{0}, c_{0}\right\rangle$ is a scalar multiple of $\mathbf{v}_{1}=\left\langle a_{1}, b_{1}, c_{1}\right\rangle$.

Determine whether the statement is true or false. Explain your answer. In these exercises $L_{0}$ and $L_{1}$ are lines in 3 -space whose parametric equations are
$$
\begin{array}{ll}{L_{0}: x=x_{0}+a_{0} t,} & {y=y_{0}+b_{0} t, \quad z=z_{0}+c_{0} t} \\ {L_{1}: x=x_{1}+a_{1} t,} & {y=y_{1}+b_{1} t, \quad z=z_{1}+c_{1} t}\end{array}
$$
If $L_{1}$ and $L_{2}$ are parallel, then $\mathbf{v}_{0}=\left\langle a_{0}, b_{0}, c_{0}\right\rangle$ is a scalar multiple of $\mathbf{v}_{1}=\left\langle a_{1}, b_{1}, c_{1}\right\rangle$.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Parametric Equations of a Line in Space

Parametric equations represent lines using one or more parameters. In 3-dimensional space, a line is usually defined by equations that express each coordinate (x, y, z) in terms of a parameter, typically denoted as t. This form clearly shows a fixed point on the line and a direction vector that indicates the line's orientation.

Direction Vectors

The direction vector of a line, extracted from the parametric equations, indicates the line's orientation in space. It is defined by the coefficients of the parameter in each coordinate (for example, <a, b, c>). This vector is central in understanding the relative orientation between lines, particularly when analyzing if two lines are parallel or intersecting.

Scalar Multiples and Parallelism

Two vectors are scalar multiples of each other if one can be obtained by multiplying the other by some nonzero constant. For lines in space, if the respective direction vectors are scalar multiples of one another, the lines are parallel because they point in the same or exactly opposite directions. This concept is fundamental when determining the parallelism of geometric objects in vector spaces.

Please give Ace some feedback