(a) Find the equations of the line through the points...

(a) Find the equations of the line through the points $(4,-1,2)$ and $(3,1,4)$. (b) Find the equation of the plane through the points $(0,0,0),(1,2,3)$ and $(2,1,1)$. (c) Find the distance from the point $(1,1,1)$ to the plane $3 x-2 y+6 z=12$. (d) Find the distance from the point $(1,0,2)$ to the line $\mathbf{r}=2 \mathbf{i}+\mathbf{i}-\mathbf{k}+(\mathbf{i}-2 \mathrm{j}+2 \mathbf{k}) t$. (e) Find the angle between the plane in (c) and the line in (d).

(a) Find the equations of the line through the points $(4,-1,2)$ and $(3,1,4)$.
(b) Find the equation of the plane through the points $(0,0,0),(1,2,3)$ and $(2,1,1)$.
(c) Find the distance from the point $(1,1,1)$ to the plane $3 x-2 y+6 z=12$.
(d) Find the distance from the point $(1,0,2)$ to the line $\mathbf{r}=2 \mathbf{i}+\mathbf{i}-\mathbf{k}+(\mathbf{i}-2 \mathrm{j}+2 \mathbf{k}) t$.
(e) Find the angle between the plane in (c) and the line in (d).

Key Concepts

Angle Between a Line and a Plane

The angle between a line and a plane is determined by finding the complementary angle of the angle between the line's direction vector and the plane's normal vector. By computing the dot product between these vectors, one can obtain the cosine of the angle between them, and then use this to find the angle between the line and the plane.

Distance from a Point to a Line in Space

This technique calculates the shortest distance from a given point to a line. It involves forming a vector from a point on the line to the external point and using the cross product with the line's direction vector. Dividing the magnitude of the resulting vector by the magnitude of the direction vector yields the perpendicular distance.

Plane Equations and Normal Vectors

A plane in three-dimensional space can be defined using three non-collinear points. By forming two vectors from these points and taking their cross product, one obtains the normal vector of the plane, which is perpendicular to every vector lying in the plane. The equation of the plane is then constructed using the point-normal form.

Vector Representation of a Line in Space

This concept involves expressing a line using an initial point and a direction vector. Given two points, one can compute the direction vector as the difference between the coordinates of these points. The line is then described by its parametric or vector equation, which allows for the representation of all the points lying on the line.

Distance from a Point to a Plane

This concept refers to the method used to determine the shortest distance between a point and a plane. The distance is found by taking the absolute value of the plane equation evaluated at the point, then dividing by the magnitude of the normal vector. This formula gives the perpendicular distance from the point to the plane.

Transcript

Please give Ace some feedback