A line has equation r⃗=a⃗+t b⃗ where r⃗=x i⃗+y j⃗+z k⃗ and a⃗ and b⃗ are constant vectors such t

step 1 All right, so looking at the parts here, for A, we have a plane perpendicular to the line and th

A line has equation r⃗=a⃗+t b⃗ where r⃗=x i⃗+y j⃗+z k⃗ and...

A line has equation $\vec{r}=\vec{a}+t \vec{b}$ where $\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}$ and $\vec{a}$ and $\vec{b}$ are constant vectors such that $\vec{a} \neq \overrightarrow{0}, \vec{b} \neq$ $\overrightarrow{0}, \vec{b}$ not parallel or perpendicular to $\vec{a} .$ For each of the planes (a)-(c), pick the equation (i)-(ix) which represents it. Explain your choice. (a) A plane perpendicular to the line and through the origin. (b) A plane perpendicular to the line and not through the origin. (c) A plane containing the line. (i) $\vec{a} \cdot \vec{r}=|| \vec{b}||$ (ii) $\vec{b} \cdot \vec{r}=|| \vec{a}||$ (iii) $\vec{a} \cdot \vec{r}=\vec{b} \cdot \vec{r}$ (iv) $(\vec{a} \times \vec{b}) \cdot(\vec{r}-\vec{a})=0$ (v) $\vec{r}-\vec{a}=\vec{b}$ (vi) $\vec{a} \cdot \vec{r}=0$ (vii) $\vec{b} \cdot \vec{r}=0$ (viii) $\vec{a}+\vec{r}=\vec{b}$ (ix) $(\vec{a} \times \vec{b}) \cdot(\vec{r}-\vec{b})=\|\vec{a}\|$

A line has equation $\vec{r}=\vec{a}+t \vec{b}$ where $\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}$ and $\vec{a}$ and $\vec{b}$ are constant vectors such that $\vec{a} \neq \overrightarrow{0}, \vec{b} \neq$ $\overrightarrow{0}, \vec{b}$ not parallel or perpendicular to $\vec{a} .$ For each of the planes (a)-(c), pick the equation (i)-(ix) which represents it. Explain your choice.
(a) A plane perpendicular to the line and through the origin.
(b) A plane perpendicular to the line and not through the origin.
(c) A plane containing the line.
(i) $\vec{a} \cdot \vec{r}=|| \vec{b}||$
(ii) $\vec{b} \cdot \vec{r}=|| \vec{a}||$
(iii) $\vec{a} \cdot \vec{r}=\vec{b} \cdot \vec{r}$
(iv) $(\vec{a} \times \vec{b}) \cdot(\vec{r}-\vec{a})=0$
(v) $\vec{r}-\vec{a}=\vec{b}$
(vi) $\vec{a} \cdot \vec{r}=0$
(vii) $\vec{b} \cdot \vec{r}=0$
(viii) $\vec{a}+\vec{r}=\vec{b}$
(ix) $(\vec{a} \times \vec{b}) \cdot(\vec{r}-\vec{b})=\|\vec{a}\|$

Key Concepts

Dot Product and Perpendicularity

The dot product is a critical operation in vector geometry that measures the cosine of the angle between two vectors. When the dot product of two vectors is zero, the vectors are orthogonal. This property is used in the plane equations to ensure that the plane is perpendicular or parallel to a given line by aligning the normal vector appropriately.

Geometric Constraints and Relative Positioning

This concept involves understanding the relationship between lines and planes in terms of position and orientation. It covers conditions such as a plane being perpendicular to a line, a plane containing a line, or a plane not passing through a particular point. Mastery of these constraints allows for the correct selection and formulation of equations that represent various geometric configurations.

Vector Equation of a Line

This concept explains how a line in space can be represented using a point and a direction vector. The equation r = a + t*b conveys that every point on the line can be generated by starting at a fixed point a and moving in the direction of vector b scaled by a parameter t. This formulation is fundamental in understanding the geometric relationships between lines and planes.

Plane Equation

A plane in space can be described by an equation involving a point on the plane and a normal vector perpendicular to the plane. One common form is n · (r - r0) = 0, where n is the normal vector and r0 is a point on the plane. This concept is crucial when determining the orientation of a plane relative to a given line, particularly when assessing conditions like perpendicularity or when ensuring the inclusion of a given line within the plane.

Cross Product and Normal Vectors

The cross product of two non-parallel vectors produces a vector that is perpendicular to both. This is used to find a suitable normal vector for a plane, especially in cases where the plane should contain or be perpendicular to a given line. Understanding the cross product is key in constructing and interpreting plane equations in three-dimensional space.

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