(a) Where does the line r⃗=2 i⃗+5 j⃗+t(3 i⃗+j⃗+2 k⃗) cut the plane x+y+z=1 ? (b) Find a vector p

step 1 So we have a three -part question here. First of all, where does this line intersect this plane?

(a) Where does the line r⃗=2 i⃗+5 j⃗+t(3 i⃗+j⃗+2 k⃗) cut the...

(a) Where does the line $\vec{r}=2 \vec{i}+5 \vec{j}+t(3 \vec{i}+\vec{j}+2 \vec{k})$ cut the plane $x+y+z=1 ?$ (b) Find a vector perpendicular to the line and lying in the plane. (c) Find an equation for the line that passes through the point of intersection of the line and plane, is perpendicular to the line, and lies in the plane.

(a) Where does the line $\vec{r}=2 \vec{i}+5 \vec{j}+t(3 \vec{i}+\vec{j}+2 \vec{k})$ cut the plane $x+y+z=1 ?$
(b) Find a vector perpendicular to the line and lying in the plane.
(c) Find an equation for the line that passes through the point of intersection of the line and plane, is perpendicular to the line, and lies in the plane.

Key Concepts

Parametric Representation of a Line

This concept refers to expressing a line in three-dimensional space using a point and a direction vector. The general form r = r0 + t*v captures every point on the line by scaling the direction vector and adding it to the fixed initial point. This form is essential for identifying specific points on the line and for subsequent algebraic manipulations, such as solving for intersections.

Equation of a Plane

A plane in 3D space can be described with a linear equation of the form ax + by + cz = d, where [a, b, c] is the normal vector to the plane. This normal vector is perpendicular to every direction within the plane, making it a crucial tool when determining intersections and ensuring perpendicularity between lines and planes.

Intersection of a Line and a Plane

To find where a line intersects a plane, one substitutes the parametric expressions of the line into the plane's equation and solves for the parameter. This operation investigates consistency between the line's path and the plane's positioning, yielding the point of intersection if one exists.

Perpendicular Vectors in a Plane

Identifying a vector that is perpendicular to a given line and lies in a specified plane involves ensuring that the vector is orthogonal to the direction vector of the line while also being contained within the plane. This often requires using conditions derived from dot products with both the line's direction and the plane's normal.

Constructing Equations of Lines in Space

When a new line must be defined subject to conditions like passing through a certain point, lying in a plane, and being perpendicular to another line, one constructs its equation using a given point and a direction vector that meets the criteria. This process might involve determining the appropriate direction vector by solving conditions based on dot-product orthogonality and confinement within the plane.

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