Given the line 𝐫=3 𝐢-𝐢+(2 𝐢+𝐣-2 𝐤) t : (a) Find the equation of the plane containing the line an

step 1 Hello everyone. Let us consider the line R is equal to 3i minus j plus 2i plus j minus 2k into t

Given the line 𝐫=3 𝐢-𝐢+(2 𝐢+𝐣-2 𝐤) t : (a) Find the equation...

Given the line $\mathbf{r}=3 \mathbf{i}-\mathbf{i}+(2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}) t$ : (a) Find the equation of the plane containing the line and the point $(2,1,0)$. (b) Find the angle between the line and the $(y, z)$ plane. (c) Find the perpendicular distance between the line and the $x$ axis. (d) Find the equation of the plane through the point $(2,1,0)$ and perpendicular to the line. (c) Find the equations of the line of intersection of the plane in $(\mathrm{d})$ and the plane $y=2 z$.

Given the line $\mathbf{r}=3 \mathbf{i}-\mathbf{i}+(2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}) t$ :
(a) Find the equation of the plane containing the line and the point $(2,1,0)$.
(b) Find the angle between the line and the $(y, z)$ plane.
(c) Find the perpendicular distance between the line and the $x$ axis.
(d) Find the equation of the plane through the point $(2,1,0)$ and perpendicular to the line.
(c) Find the equations of the line of intersection of the plane in $(\mathrm{d})$ and the plane $y=2 z$.

Key Concepts

Parametric Equations of a Line in 3D

The parametric representation of a line in three-dimensional space expresses the line as a combination of a fixed point and a direction vector scaled by a parameter. This form is foundational in analytic geometry as it allows easy computation of points on the line and facilitates further operations, such as finding intersections or distances with other geometric entities.

Plane Equation in 3D

A plane in three-dimensional space can be represented by a linear equation in x, y, and z. Typically, the point-normal form of a plane’s equation is used, where a fixed point on the plane and a normal vector (perpendicular to the plane) define the plane. This representation is essential for solving intersection problems and determining orientations between geometric objects.

Determining a Plane Through a Line and a Point

To find a plane that contains a given line and an additional point, one uses the fact that the line’s direction vector lies in the plane and forms a second vector by connecting any point on the line to the given external point. The cross product of these two vectors yields the normal vector to the plane, which can then be used with the point-normal form to obtain the plane’s equation.

Angle Between a Line and a Plane or Coordinate Plane

The angle between a line and a plane (or a coordinate plane) is often found by relating the line's direction vector to the plane’s normal vector. Alternatively, when dealing with coordinate planes, one may consider the projection of the line onto the plane and use trigonometric relationships to find the acute angle between the line and the plane. This concept is useful for analyzing the orientation of lines relative to given planes.

Distance Between a Line and a Coordinate Axis

Computing the shortest or perpendicular distance between a line and a coordinate axis (or between skew lines) involves minimizing the distance between any point on the line and any point on the axis. This typically requires forming a vector between a point on the line and the axis, and then using projections and the concept of orthogonality to find the minimum distance.

Plane Perpendicular to a Line

A plane that is perpendicular to a given line has a normal vector that is parallel to the direction vector of the line. Given a point through which the plane passes, one can directly use the direction vector of the line as the normal in the point-normal equation of the plane. This concept is particularly useful in problems involving orthogonal projections and perpendicular distances.

Intersection of Two Planes

The line of intersection of two planes is the set of points that simultaneously satisfy both plane equations. To find this line, one typically solves the two equations as a system of linear equations in three variables. The resulting parametric equations describe the line along which the planes intersect, highlighting the interplay between linear algebra and geometry.

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