Question

A pair of lines in ${\mathbb{R}}^3$ are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect, determine the point(s) of intersection. $\mathbf{r}(t)=⟨1+2t,7-3t,6+t⟩;$ $\mathbf{R}(s)=⟨-9+6s,22-9s,1+3s⟩$

Short answer

If they intersect, find the point of intersection. Given lines: $\mathbf{r}(t)=⟨1+2t,7-3t,6+t⟩$ $\mathbf{R}(s)=⟨-9+6s,22-9s,1+3s⟩$ Answer: The given lines are coplanar and parallel. Since parallel lines do not intersect, there is no point of intersection.

Step-by-step solution

Identify the line equations

The problem provides us with two line equations: $\mathbf{r}(t)=⟨1+2t,7-3t,6+t⟩;$ $\mathbf{R}(s)=⟨-9+6s,22-9s,1+3s⟩$

Write the point-vector equations of the lines.

The point-vector equations of the lines are: $\mathbf{r}(t)=(1,7,6)+t(2,-3,1)$ $\mathbf{R}(s)=(-9,22,1)+s(6,-9,3)$

Determine if lines are coplanar

Let $\mathbf{A}=(1,7,6)$ and $\mathbf{B}=(-9,22,1)$ be the points on the lines, and let ${\mathbf{v}}_1=(2,-3,1)$ and ${\mathbf{v}}_2=(6,-9,3)$ be the direction vectors of the lines. Calculate $\mathbf{C}=\mathbf{B}-\mathbf{A}$: $\mathbf{C}=(-10,15,-5)$ Calculate the cross product of the direction vectors, ${\mathbf{v}}_1\times {\mathbf{v}}_2$: ${\mathbf{v}}_1\times {\mathbf{v}}_2=(0,0,0)$ Now, calculate the dot product of $\mathbf{C}$ and ${\mathbf{v}}_1\times {\mathbf{v}}_2$: $\mathbf{C}\cdot ({\mathbf{v}}_1\times {\mathbf{v}}_2)=0$ Since the dot product is zero, the lines are coplanar, which means they are either parallel or intersecting.

Determine if lines are parallel

If the direction vectors are scalar multiples of each other, then the lines are parallel. Since ${\mathbf{v}}_2=3{\mathbf{v}}_1$, the lines are parallel.

Conclusion

The given pair of lines are parallel. Since there are no points of intersection for parallel lines, there is no need to find the point(s) of intersection.

Key concepts

Vector Equations

Vector equations are a powerful tool for describing lines in three-dimensional space. A vector equation of a line is typically expressed in the form $\mathbf{r}(t)=\mathbf{a}+t\mathbf{v}$, where:

• $\mathbf{a}$ is a position vector, representing a point on the line.
• $t$ is a parameter, usually real number, that varies.
• $\mathbf{v}$ is a direction vector, which indicates the direction in which the line extends.

This form allows us to easily locate any point on the line by adjusting the parameter $t$. It is essential for solving problems involving lines in three dimensions, such as determining intersections or assessing parallelism.

Direction Vectors

Direction vectors are crucial in understanding the orientation of a line in space. For a line given by a vector equation $\mathbf{r}(t)=\mathbf{a}+t\mathbf{v}$, the vector $\mathbf{v}$ is the direction vector. This vector:

• Shows the line's direction.
• Is parallel to the line.
• Helps in determining if two lines are parallel. If two lines have direction vectors that are scalar multiples of each other, they are parallel.

Direction vectors give a clear understanding of how a line "moves" through space, and they are also used in cross product calculations to determine the skewness or intersection of lines.

Cross Product

The cross product is a vector operation used to find a vector perpendicular to two given vectors. For vectors $\mathbf{a}=(a_1,a_2,a_3)$ and $\mathbf{b}=(b_1,b_2,b_3)$, the cross product $\mathbf{a}\times \mathbf{b}$ is calculated as follows:$$\mathbf{a}\times \mathbf{b}=(a_2{b}_3-a_3{b}_2,a_3{b}_1-a_1{b}_3,a_1{b}_2-a_2{b}_1)$$This result is a new vector that is orthogonal to both $\mathbf{a}$ and $\mathbf{b}$. In the context of lines, if the cross product of two direction vectors is zero, the vectors are parallel, and hence the lines may be coplanar. Otherwise, they are skew.

Dot Product

The dot product, also known as a scalar product, measures the extent to which two vectors point in the same direction. For vectors $\mathbf{a}=(a_1,a_2,a_3)$ and $\mathbf{b}=(b_1,b_2,b_3)$, the dot product is:$$\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$$This operation results in a scalar value. In determining coplanarity, if the vector $\mathbf{C}$, which is the difference between two points on the lines, has a zero dot product with the cross product of the direction vectors, then the lines might be coplanar. This value offers insights into the relative orientation of the vectors involved, facilitating problem-solving with lines in ${\mathbb{R}}^3$.