IB Mathematics Analysis and Approaches Topic AHL 3.17—vector Equations of a Plane Revision Notes (HL)

IB Mathematics Analysis and Approaches Topic AHL 3.17—vector Equations of a Plane Revision Notes (HL) | RevisionDojo

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Vector Equations of a Plane

Parametric Form: r = a + λb + μc

The vector equation of a plane in its parametric form is given by:

$r = a + λb + μc$

Where:

$r$ is the position vector of any point on the plane
$a$ is the position vector of a fixed point on the plane
$b$ and $c$ are non-parallel vectors lying within the plane
$λ$ and $μ$ are scalar parameters

This equation represents all points on the plane as a linear combination of two non-parallel vectors ($b$ and $c$) added to a fixed point ($a$).

Note

The vectors $b$ and $c$ must be non-parallel to ensure they span the entire plane.

Interpretation

The vector $a$ anchors the plane at a specific point in space.
Vectors $b$ and $c$ define the "directions" in which the plane extends.
By varying $λ$ and $μ$, we can reach any point on the plane.

Example

Let $\mathbf{a} = (1,2,3) $, $\mathbf{b} = (1,0,1) $, and $\mathbf{c} = (0,1,-1) $. The equation of the plane is: $$\mathbf{r} = (1,2,3) + \lambda(1,0,1) + \mu(0,1,-1).$$ This can be written in component form as: $$x = 1 + \lambda,\ y = 2 + \mu,\ z = 3 + \lambda - \mu.$$

Hint

Understanding Vector Equations: Lines vs. Planes Planes

A vector line equation extends a point one-dimensionally using a single direction vector, allowing movement along that direction.
In contrast, a vector plane equation extends a point two-dimensionally using two direction vectors, defining a surface rather than a single path.

Normal Form: (r-a) • n = 0

Another way to represent a plane is using its normal vector:

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Vector Equations of a Plane

Parametric Form: r = a + λb + μc

The vector equation of a plane in its parametric form is given by:

$r = a + λb + μc$

Where:

$r$ is the position vector of any point on the plane
$a$ is the position vector of a fixed point on the plane
$b$ and $c$ are non-parallel vectors lying within the plane
$λ$ and $μ$ are scalar parameters

This equation represents all points on the plane as a linear combination of two non-parallel vectors ($b$ and $c$) added to a fixed point ($a$).

Note

The vectors $b$ and $c$ must be non-parallel to ensure they span the entire plane.

Interpretation

The vector $a$ anchors the plane at a specific point in space.
Vectors $b$ and $c$ define the "directions" in which the plane extends.
By varying $λ$ and $μ$, we can reach any point on the plane.

Example

Hint

Understanding Vector Equations: Lines vs. Planes Planes

A vector line equation extends a point one-dimensionally using a single direction vector, allowing movement along that direction.
In contrast, a vector plane equation extends a point two-dimensionally using two direction vectors, defining a surface rather than a single path.

Normal Form: (r-a) • n = 0

Another way to represent a plane is using its normal vector:

Definition

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Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

IB Mathematics Analysis and Approaches AHL 3.17—vector Equations of a Plane Notes

Vector Equations of a Plane

Parametric Form: r = a + λb + μc

Interpretation

Understanding Vector Equations: Lines vs. Planes Planes

Normal Form: (r-a) • n = 0

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Vector Equations of a Plane

Parametric Form: r = a + λb + μc

Interpretation

Understanding Vector Equations: Lines vs. Planes Planes

Normal Form: (r-a) • n = 0

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics