2WBB0Calculus I

2WBB0 · Topic 02

Vector geometry track

Angles, lines, planes, and distances for programmes assigned the vector branch.

What you need to be able to do

Move between parametric and normal descriptions of lines and planes.
Use dot and cross products for angles, normals, and intersection directions.
Compute point-to-line and point-to-plane distances reliably.

1. Dot product, cross product, and angle

The dot product detects perpendicularity and gives angles. The cross product produces a vector perpendicular to two three-dimensional vectors.

An exam may hide the needed direction vector in a line equation. For y=mx+b, a direction vector is (1,m), while a perpendicular direction is (m,-1) or (-m,1).

Exam method

Extract or construct direction vectors before applying any formula.
Use a dot product equal to zero for perpendicularity.
Normalize only when the question needs a unit vector.
For an angle, check whether the question wants the acute angle or the oriented angle.

Dot product, cross product, and angle: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Dot product, cross product, and angleSelf-review

Worked example 1

a=(1,2)

2. Lines and planes from equations

A line in three dimensions needs a support point and one direction vector. A plane needs a support point and a normal, or a support point with two independent directions.

The intersection of two nonparallel planes is a line. Its direction is perpendicular to both plane normals, so a cross product gives it immediately.

Exam method

For two plane equations, cross their normals to get the line direction.
Set one coordinate as a parameter and solve the two equations for the others.
Check the support point in every original equation.
Keep support and direction vectors visually separate in the final representation.

Lines and planes from equations: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Lines and planes from equationsSelf-review

Worked example 1

x+y+z=-1,\qquad x+2y+3z=-4.

3. Distances to lines and planes

Distance is measured along a perpendicular. For a plane, project the point displacement onto the plane normal. For a line, either use the cross-product distance formula or find the perpendicular foot.

When a line is given as two plane equations, first convert it to parametric form. Most lost marks come from using the wrong direction vector rather than from the distance calculation.

Exam method

Identify a support point and direction or normal vector.
Form the displacement from the support point to the given point.
Apply the matching distance formula and keep the absolute value.
Check that the result is nonnegative and has sensible scale.

Distances to lines and planes: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Distances to lines and planesSelf-review

Worked example 1

P=(7,-1,4)

Formulas used in this topic

$a\cdot b=\sum_i a_i b_i,\qquad \cos\theta=\frac{a\cdot b}{\|a\|\|b\|}$
$a\times b=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$
$d(P,\Pi)=\frac{|n\cdot(P-P_0)|}{\|n\|}$
$d(P,\ell)=\frac{\|(P-P_0)\times v\|}{\|v\|}$

2WBB0 · Topic 02

Vector geometry track

Angles, lines, planes, and distances for programmes assigned the vector branch.

What you need to be able to do

Move between parametric and normal descriptions of lines and planes.
Use dot and cross products for angles, normals, and intersection directions.
Compute point-to-line and point-to-plane distances reliably.

1. Dot product, cross product, and angle

The dot product detects perpendicularity and gives angles. The cross product produces a vector perpendicular to two three-dimensional vectors.

An exam may hide the needed direction vector in a line equation. For y=mx+b, a direction vector is (1,m), while a perpendicular direction is (m,-1) or (-m,1).

Exam method

Extract or construct direction vectors before applying any formula.
Use a dot product equal to zero for perpendicularity.
Normalize only when the question needs a unit vector.
For an angle, check whether the question wants the acute angle or the oriented angle.

Dot product, cross product, and angle: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Dot product, cross product, and angleSelf-review

Worked example 1

a=(1,2)

2. Lines and planes from equations

A line in three dimensions needs a support point and one direction vector. A plane needs a support point and a normal, or a support point with two independent directions.

The intersection of two nonparallel planes is a line. Its direction is perpendicular to both plane normals, so a cross product gives it immediately.

Exam method

For two plane equations, cross their normals to get the line direction.
Set one coordinate as a parameter and solve the two equations for the others.
Check the support point in every original equation.
Keep support and direction vectors visually separate in the final representation.

Lines and planes from equations: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Lines and planes from equationsSelf-review

Worked example 1

x+y+z=-1,\qquad x+2y+3z=-4.

3. Distances to lines and planes

When a line is given as two plane equations, first convert it to parametric form. Most lost marks come from using the wrong direction vector rather than from the distance calculation.

Exam method

Identify a support point and direction or normal vector.
Form the displacement from the support point to the given point.
Apply the matching distance formula and keep the absolute value.
Check that the result is nonnegative and has sensible scale.

Distances to lines and planes: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Distances to lines and planesSelf-review

Worked example 1

P=(7,-1,4)

Formulas used in this topic

$a\cdot b=\sum_i a_i b_i,\qquad \cos\theta=\frac{a\cdot b}{\|a\|\|b\|}$
$a\times b=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$
$d(P,\Pi)=\frac{|n\cdot(P-P_0)|}{\|n\|}$
$d(P,\ell)=\frac{\|(P-P_0)\times v\|}{\|v\|}$

2WBB0 · Topic 02

Vector geometry track

Angles, lines, planes, and distances for programmes assigned the vector branch.

What you need to be able to do

Move between parametric and normal descriptions of lines and planes.
Use dot and cross products for angles, normals, and intersection directions.
Compute point-to-line and point-to-plane distances reliably.

1. Dot product, cross product, and angle

The dot product detects perpendicularity and gives angles. The cross product produces a vector perpendicular to two three-dimensional vectors.

An exam may hide the needed direction vector in a line equation. For y=mx+b, a direction vector is (1,m), while a perpendicular direction is (m,-1) or (-m,1).

Exam method

Extract or construct direction vectors before applying any formula.
Use a dot product equal to zero for perpendicularity.
Normalize only when the question needs a unit vector.
For an angle, check whether the question wants the acute angle or the oriented angle.

Dot product, cross product, and angle: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Dot product, cross product, and angleSelf-review

Worked example 1

a=(1,2)

2. Lines and planes from equations

A line in three dimensions needs a support point and one direction vector. A plane needs a support point and a normal, or a support point with two independent directions.

The intersection of two nonparallel planes is a line. Its direction is perpendicular to both plane normals, so a cross product gives it immediately.

Exam method

For two plane equations, cross their normals to get the line direction.
Set one coordinate as a parameter and solve the two equations for the others.
Check the support point in every original equation.
Keep support and direction vectors visually separate in the final representation.

Lines and planes from equations: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Lines and planes from equationsSelf-review

Worked example 1

x+y+z=-1,\qquad x+2y+3z=-4.

3. Distances to lines and planes

When a line is given as two plane equations, first convert it to parametric form. Most lost marks come from using the wrong direction vector rather than from the distance calculation.

Exam method

Identify a support point and direction or normal vector.
Form the displacement from the support point to the given point.
Apply the matching distance formula and keep the absolute value.
Check that the result is nonnegative and has sensible scale.

Distances to lines and planes: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Distances to lines and planesSelf-review

Worked example 1

P=(7,-1,4)

Formulas used in this topic

$a\cdot b=\sum_i a_i b_i,\qquad \cos\theta=\frac{a\cdot b}{\|a\|\|b\|}$
$a\times b=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$
$d(P,\Pi)=\frac{|n\cdot(P-P_0)|}{\|n\|}$
$d(P,\ell)=\frac{\|(P-P_0)\times v\|}{\|v\|}$