§4 Vectors and Geometry

1. The Big Idea

Vectors are where algebra and geometry start speaking the same language.

A vector can be understood as:

• an arrow with length and direction
• a coordinate list, like $(3,-1,4)$
• a displacement from one point to another
• a tool for describing lines, planes, angles, areas, volumes, and distances

The main habit is this: translate geometric words into vector operations.

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2. Vector Basics

A vector is an object with magnitude and direction.

You can picture it as an arrow. The tail is the starting point, and the tip is the ending point.

If a vector starts at point $A$ and ends at point $B$, we write

$$\vec{AB}$$

Coordinate Form

In $\mathbb R^2$, a vector has two components:

$$\vec{u}=(u_1,u_2)$$

In $\mathbb R^3$, a vector has three components:

$$\vec{u}=(u_1,u_2,u_3)$$

In $\mathbb R^n$, a vector has $n$ components:

$$\vec{u}=(u_1,u_2,\dots,u_n)$$

Vector from One Point to Another

To find the vector from $A$ to $B$, subtract coordinates in this order:

$$\vec{AB}=B-A$$

In $\mathbb R^3$, if

$$A=(x_1,y_1,z_1)$$

and

$$B=(x_2,y_2,z_2)$$

then

$$\vec{AB}=(x_2-x_1,\ y_2-y_1,\ z_2-z_1)$$

The shortcut phrase is: terminal minus initial.

Example

If

$$A=(1,4,-2),\qquad B=(5,1,3)$$

then

$$\vec{AB}=(5-1,\ 1-4,\ 3-(-2))=(4,-3,5)$$

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3. Vector Addition and Scalar Multiplication

Vector addition and subtraction happen component-by-component.

If

$$\vec{u}=(u_1,u_2,\dots,u_n)$$

and

$$\vec{v}=(v_1,v_2,\dots,v_n)$$

then

$$\vec{u}+\vec{v}=(u_1+v_1,u_2+v_2,\dots,u_n+v_n)$$

and

$$\vec{u}-\vec{v}=(u_1-v_1,u_2-v_2,\dots,u_n-v_n)$$

For a scalar $k$,

$$k\vec{u}=(ku_1,ku_2,\dots,ku_n)$$

Geometrically, adding vectors combines displacements. Multiplying by a scalar stretches, shrinks, or reverses a vector.

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4. Length, Unit Vectors, and Parallel Vectors

Norm

The norm of a vector is its length.

For

$$\vec{u}=(u_1,u_2,\dots,u_n)$$

the norm is

$$|\vec{u}|=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}$$

Example:

$$\vec{u}=(3,4)$$

Then

$$|\vec{u}|=\sqrt{3^2+4^2}=5$$

Unit Vectors

A unit vector has length $1$.

To make a unit vector in the same direction as $\vec{u}$, divide by the length of $\vec{u}$:

$$\frac{\vec{u}}{|\vec{u}|}$$

This keeps the direction but removes the length.

If you want the opposite direction, use

$$-\frac{\vec{u}}{|\vec{u}|}$$

Parallel Vectors

Two nonzero vectors $\vec{u}$ and $\vec{v}$ are parallel if one is a scalar multiple of the other:

$$\vec{u}=k\vec{v}$$

for some scalar $k$.

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5. Dot Product: Angles and Perpendicularity

The dot product combines two vectors and gives a number.

For

$$\vec{u}=(u_1,u_2,\dots,u_n)$$

and

$$\vec{v}=(v_1,v_2,\dots,v_n)$$

the dot product is

$$\vec{u}\cdot\vec{v}=u_1v_1+u_2v_2+\cdots+u_nv_n$$

The dot product is useful because it also knows about angles:

$$\vec{u}\cdot\vec{v}=|\vec{u}||\vec{v}|\cos\theta$$

where $\theta$ is the angle between the vectors.

So

$$\cos\theta=\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}$$

Reading the Sign of a Dot Product

Orthogonal Vectors

Two vectors are orthogonal if they meet at a right angle.

Algebraically:

$$\vec{u}\cdot\vec{v}=0$$

So dot product zero means perpendicular.

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6. Distance in $\mathbb R^n$

Distance between two points is the length of the displacement vector between them.

In vector language:

$$d(A,B)=|\vec{AB}|$$

If

$$A=(a_1,a_2,\dots,a_n)$$

and

$$B=(b_1,b_2,\dots,b_n)$$

then

$$d(A,B)=\sqrt{(b_1-a_1)^2+(b_2-a_2)^2+\cdots+(b_n-a_n)^2}$$

This is just the norm formula applied to $\vec{AB}$.

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7. Projection

Projection means the “shadow” of one vector in the direction of another vector.

The projection of $\vec{u}$ onto $\vec{a}$ is

$$\operatorname{proj}_{\vec{a}}\vec{u} = \frac{\vec{u}\cdot\vec{a}}{|\vec{a}|^2}\vec{a}$$

This gives the component of $\vec{u}$ that points in the direction of $\vec{a}$.

The perpendicular component is what is left over:

$$\vec{u}-\operatorname{proj}_{\vec{a}}\vec{u}$$

Projection is useful for:

• splitting a vector into parallel and perpendicular parts
• closest point problems
• distance problems
• understanding shadows/components in a chosen direction

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8. Lines and Planes: Main Ingredients

Lines and planes are built from points and direction information.

A direction vector lies along the object.

A normal vector is perpendicular to the object.

This distinction is the key to most geometry problems.

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9. Lines in $\mathbb R^2$: Point-Normal Form

A line in $\mathbb R^2$ can be described using a point and a normal vector.

If the line passes through

$$A(x_0,y_0)$$

and has normal vector

$$\vec{n}=(a,b)$$

then the equation is

$$a(x-x_0)+b(y-y_0)=0$$

This can be rewritten as

$$ax+by+c=0$$

The normal vector $\vec{n}=(a,b)$ is perpendicular to the line. That is why it appears in the equation.

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10. Planes in $\mathbb R^3$: Point-Normal Form

A plane in $\mathbb R^3$ can also be described using a point and a normal vector.

If the plane passes through

$$A(x_0,y_0,z_0)$$

and has normal vector

$$\vec{n}=(a,b,c)$$

then its equation is

$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$

This can be rewritten as

$$ax+by+cz+d=0$$

The normal vector controls the orientation of the plane. If you know a point on the plane and a vector perpendicular to the plane, you know the plane.

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11. Parametric Lines and Planes

Parametric equations use one or more parameters as sliders.

Line in $\mathbb R^3$

A line needs:

1. a point
2. a direction vector

If a line passes through

$$A(x_0,y_0,z_0)$$

and is parallel to

$$\vec{v}=(a,b,c)$$

then

$$(x,y,z)=(x_0,y_0,z_0)+t(a,b,c)$$

So

$$x=x_0+at$$ $$y=y_0+bt$$ $$z=z_0+ct$$

The parameter $t$ moves you along the line.

Symmetric Equation of a Line

If $a,b,c\neq 0$, then

$$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$$

This is the symmetric form.

Parametric Plane

A plane in parametric form needs:

1. a point
2. two nonparallel direction vectors

If the plane passes through

$$A(x_0,y_0,z_0)$$

and is parallel to

$$\vec{v}_1=(a_1,b_1,c_1)$$

and

$$\vec{v}_2=(a_2,b_2,c_2)$$

then

$$(x,y,z) = (x_0,y_0,z_0) + s\vec{v}_1 + t\vec{v}_2$$

In components:

$$x=x_0+a_1s+a_2t$$ $$y=y_0+b_1s+b_2t$$ $$z=z_0+c_1s+c_2t$$

The parameters $s$ and $t$ let you move in two independent directions across the plane.

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12. Cross Product

The cross product is defined only in $\mathbb R^3$.

For

$$\vec{u}=(u_1,u_2,u_3)$$

and

$$\vec{v}=(v_1,v_2,v_3)$$

the cross product is

$$\vec{u}\times\vec{v} = (u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1)$$

The output is a vector, not a number.

That vector is perpendicular to both $\vec{u}$ and $\vec{v}$.

Cross Product Properties

$$\vec{u}\times\vec{u}=\vec{0}$$ $$\vec{u}\times\vec{v}=-(\vec{v}\times\vec{u})$$ $$\vec{u}\cdot(\vec{u}\times\vec{v})=0$$ $$\vec{v}\cdot(\vec{u}\times\vec{v})=0$$

If $\vec{u}$ and $\vec{v}$ are parallel, then

$$\vec{u}\times\vec{v}=\vec{0}$$

So the cross product is useful when you need a vector perpendicular to two given vectors.

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13. Area and Volume

The cross product and scalar triple product turn geometry into formulas.

Area Using Cross Product

The area of the parallelogram determined by $\vec{u}$ and $\vec{v}$ is

$$|\vec{u}\times\vec{v}|$$

The area of the triangle determined by $\vec{u}$ and $\vec{v}$ is

$$\frac12|\vec{u}\times\vec{v}|$$

If the triangle has points $A,B,C$, use

$$\vec{AB}$$

and

$$\vec{AC}$$

Then

$$\text{Area}=\frac12|\vec{AB}\times\vec{AC}|$$

Scalar Triple Product

For vectors $\vec{u},\vec{v},\vec{w}$ in $\mathbb R^3$, the scalar triple product is

$$\vec{u}\cdot(\vec{v}\times\vec{w})$$

It can be computed as the determinant

$$\begin{vmatrix} u_1&u_2&u_3\\ v_1&v_2&v_3\\ w_1&w_2&w_3 \end{vmatrix}$$

Volume of a Parallelepiped

The volume of the parallelepiped determined by $\vec{u},\vec{v},\vec{w}$ is

$$|\vec{u}\cdot(\vec{v}\times\vec{w})|$$

If the scalar triple product is zero, then the vectors lie in the same plane. That means they are coplanar.

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14. Distance Formulas

Distance formulas usually come from the same idea: the shortest distance is perpendicular.

Distance from a Point to a Plane

For plane

$$ax+by+cz+d=0$$

and point

$$P(x_0,y_0,z_0)$$

the distance from $P$ to the plane is

$$d= \frac{|ax_0+by_0+cz_0+d|} {\sqrt{a^2+b^2+c^2}}$$

The numerator measures how far the point is from satisfying the plane equation. The denominator normalizes by the length of the normal vector.

Distance from a Point to a Line in $\mathbb R^3$

Suppose a line passes through point $A$ and has direction vector $\vec{v}$.

Let $P$ be another point. Then

$$d= \frac{|\vec{v}\times\vec{AP}|}{|\vec{v}|}$$

Why? The cross product gives the area of the parallelogram formed by $\vec{v}$ and $\vec{AP}$.

That area is also

$$\text{base}\times\text{height}$$

The base is $|\vec{v}|$, and the height is the distance.

Distance Between Skew Lines

Suppose line $L_1$ passes through $A_1$ with direction $\vec{v}_1$, and line $L_2$ passes through $A_2$ with direction $\vec{v}_2$.

First create a vector perpendicular to both lines:

$$\vec{n}=\vec{v}_1\times\vec{v}_2$$

Then the distance between the skew lines is

$$d= \frac{|\vec{A_1A_2}\cdot\vec{n}|}{|\vec{n}|}$$

This measures the component of the connecting vector in the direction perpendicular to both lines.

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15. Relationships Between Lines and Planes

These relationship tests are common in geometry problems.

Line and Plane

Suppose a line has direction vector $\vec{v}$, and a plane has normal vector $\vec{n}$.

Two Lines in $\mathbb R^3$

Let two lines have direction vectors $\vec{v}_1$ and $\vec{v}_2$.

Skew lines exist only in 3D. They miss each other without being parallel.

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16. Closest Point Problems

Closest point problems are really perpendicularity problems.

Closest Point on a Plane

To find the closest point on a plane to a point $A$:

1. Find the normal vector $\vec{n}$ of the plane.
2. Create a line through $A$ in direction $\vec{n}$.
3. Intersect that line with the plane.
4. The intersection point is the closest point.

Why? The shortest path from a point to a plane is perpendicular to the plane.

Closest Point on a Line

To find the closest point on a line to point $A$:

1. Use the line direction vector $\vec{v}$.
2. Create a plane through $A$ whose normal vector is $\vec{v}$.
3. Intersect that plane with the line.
4. The intersection point is the closest point.

Why? The shortest segment from a point to a line is perpendicular to the line.

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17. Plane Construction Recipes

Many plane problems are about finding a normal vector. Once you have the normal vector and a point, use point-normal form.

Plane Through Three Points

Given three points $A,B,C$:

1. Compute two direction vectors:

$$\vec{AB}$$

and

$$\vec{AC}$$

2. Cross them to get a normal vector:

$$\vec{n}=\vec{AB}\times\vec{AC}$$

3. Use point-normal form:

$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$

Plane Containing a Line and a Point

If a plane contains a line and a point not on the line:

1. Use the direction vector of the line.
2. Use a vector from a point on the line to the extra point.
3. Cross those two vectors to get the normal vector.
4. Use point-normal form.

Plane Containing Two Parallel Lines

If two lines are parallel:

1. Use their common direction vector.
2. Use a vector connecting a point on the first line to a point on the second line.
3. Cross those two vectors to get the plane normal.
4. Use point-normal form.

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18. Problem-Solving Routine

For vector geometry problems, use this routine:

1. Identify the goal. Are you finding a vector, length, angle, line, plane, area, volume, or distance?
2. Write down the given points and vectors. Keep labels clear.
3. If you need a direction vector, subtract points. Use terminal minus initial.
4. If you need perpendicularity, use dot product or cross product. Dot product tests perpendicularity; cross product creates perpendicular vectors in $\mathbb R^3$.
5. If you need a plane, look for a normal vector. A normal plus a point gives point-normal form.
6. If you need a distance, choose the right formula. Point-plane, point-line, and skew-line distances use different setups.
7. Check the type of answer. A distance is a scalar, a direction is a vector, a line/plane is an equation, and an intersection is a point.

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19. Key Takeaways

• A vector has magnitude and direction.
• To find $\vec{AB}$, compute terminal minus initial.
• The norm gives vector length.
• A unit vector keeps direction but has length $1$.
• Dot product gives angle information and detects perpendicularity.
• Projection gives the shadow of one vector in another direction.
• A line needs a point and direction vector.
• A plane in point-normal form needs a point and normal vector.
• Cross product creates a vector perpendicular to two vectors in $\mathbb R^3$.
• Cross product gives area; scalar triple product gives volume.
• Shortest distance problems usually involve perpendicular directions.

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Mini-Self-Check

1. If $A=(2,-1,4)$ and $B=(5,3,0)$, what is $\vec{AB}$?

2. What does $\vec{u}\cdot\vec{v}=0$ tell you?

3. What does $\vec{u}\times\vec{v}$ give you?

4. What do you need to write a line in $\mathbb R^3$?

5. What do you need to write a plane in point-normal form?