LEARN · EXPLAIN · REVISE

Space vectors and solid geometry: mixed review

Read the idea, work independently, then explain what changed.

TOPIC 01

Space vectors and solid geometry: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Find the first component of a+b.

a=(20,2,-1),\quad b=(3,-2,4)

The angle formula requires two nonzero vectors; a dot product is a scalar.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

(a+b)_1=a_1+b_1

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$(a+b)=(23,0,3)$
Cancellation of other components does not affect the first component.

The requested value is 23.

Checks and common pitfalls: Cancellation of other components does not affect the first component.

Think first. Reveal a hint when the class is ready.

02 / Foundation#Your turn

Find the coefficient of e1 when e1=(1,0,0), e2=(1,1,0), e3=(0,0,1).

v=(23,2,3)

A basis must be independent. Weights representing a point in a plane sum to one.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

(x+y,y,z)=v

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$y=2,\quad z=3,\quad x=23-2=21$
Changing the basis changes coefficients even when the geometric vector is unchanged.

The requested value is 21.

Checks and common pitfalls: Changing the basis changes coefficients even when the geometric vector is unchanged.

Think first. Reveal a hint when the class is ready.

03 / Foundation#Your turn

Find λ so that a and b are perpendicular.

a=(1,2,3),\ b=(22,1,\lambda)

Projection requires a nonzero direction; signed scalar projection can be negative.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the dot product to connect coordinates with length or angle.

Hint 2

Use this intermediate relation.

a\cdot b=0

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$22+2+3\lambda=0$
Apply the stated relation and retain its conditions.
$\lambda=-24/3$
Use a zero dot product, not componentwise multiplication equal to zero.

The requested value is -8.

Checks and common pitfalls: Use a zero dot product, not componentwise multiplication equal to zero.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Find the distance from P to the plane.

P=(0,0,23),\quad \Pi:z=2

Use acute line/plane angles; normals must be nonzero and distances nonnegative.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the dot product to connect coordinates with length or angle.

Hint 2

Use this intermediate relation.

d=|z_P-2|

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$d=|23-2|=21$
The normal displacement gives the shortest distance.

The requested value is 21.

Checks and common pitfalls: The normal displacement gives the shortest distance.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Find the first component of a+b.

a=(24,2,-1),\quad b=(3,-2,4)

The angle formula requires two nonzero vectors; a dot product is a scalar.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

(a+b)_1=a_1+b_1

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$(a+b)=(27,0,3)$
Cancellation of other components does not affect the first component.

The requested value is 27.

Checks and common pitfalls: Cancellation of other components does not affect the first component.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Find the coefficient of e1 when e1=(1,0,0), e2=(1,1,0), e3=(0,0,1).

v=(27,2,3)

A basis must be independent. Weights representing a point in a plane sum to one.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

(x+y,y,z)=v

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$y=2,\quad z=3,\quad x=27-2=25$
Changing the basis changes coefficients even when the geometric vector is unchanged.

The requested value is 25.

Checks and common pitfalls: Changing the basis changes coefficients even when the geometric vector is unchanged.

Think first. Reveal a hint when the class is ready.

07 / Standard#Your turn

Find cos θ between a and b.

a=(1,1,0), b=(0,26,1)

Projection requires a nonzero direction; signed scalar projection can be negative.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the dot product to connect coordinates with length or angle.

Hint 2

Use this intermediate relation.

cosθ=(a·b)/(|a||b|)

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$a\cdot b=26, |a|=\sqrt2, |b|=\sqrt{677}$
Apply the stated relation and retain its conditions.
$\cos\theta=26/\sqrt{1354}$
Both vector lengths enter the denominator.

The requested value is 0.706584352758.

Checks and common pitfalls: Both vector lengths enter the denominator.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Find the distance from P to the plane.

P=(27,0,0),\quad \Pi:3x+4y=0

Use acute line/plane angles; normals must be nonzero and distances nonnegative.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the dot product to connect coordinates with length or angle.

Hint 2

Use this intermediate relation.

d=|3x_P+4y_P|/5

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$d=3(27)/\sqrt{3^2+4^2}=16.2$
Normalize the plane normal; raw substitution is not a distance.

The requested value is 16.2.

Checks and common pitfalls: Normalize the plane normal; raw substitution is not a distance.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Are a and b parallel? Justify your answer.

a=(1,2,28),\quad b=(2,4,56)

The angle formula requires two nonzero vectors; a dot product is a scalar.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

b=\lambda a

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$(2,4,56)=2(1,2,28)$
All components have the same scalar multiplier.

The requested relation or conclusion is shown below.

b=2a

Checks and common pitfalls: All components have the same scalar multiplier.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Find λ so that the third vector lies in the plane spanned by the first two.

a=(1,0,0),\ b=(0,1,0),\ c=(29,2,\lambda)

A basis must be independent. Weights representing a point in a plane sum to one.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

c=xa+yb

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$c=(x,y,0)\Rightarrow\lambda=0$
The plane spanned by a and b has zero third coordinate.

The requested value is 0.

Checks and common pitfalls: The plane spanned by a and b has zero third coordinate.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Find the signed scalar projection of v onto u.

v=(30,2,0),\ u=(1,0,0)

Projection requires a nonzero direction; signed scalar projection can be negative.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the dot product to connect coordinates with length or angle.

Hint 2

Use this intermediate relation.

s=v\cdot u/|u|

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$s=30/1=30$
A unit coordinate direction picks out the matching component.

The requested value is 30.

Checks and common pitfalls: A unit coordinate direction picks out the matching component.

Think first. Reveal a hint when the class is ready.

12 / Standard#Your turn

Does the entire line lie in the plane?

X=(0,0,31)+s(1,-1,0), \Pi:x+y+z=31

Use acute line/plane angles; normals must be nonzero and distances nonnegative.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the dot product to connect coordinates with length or angle.

Hint 2

n·u=0 and the initial point is in the plane.

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$x+y+z=s-s+31=31$
Direction parallelism plus an incident point proves containment.

The requested relation or conclusion is shown below.

\ell\subset\Pi

Checks and common pitfalls: Direction parallelism plus an incident point proves containment.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Are a and b parallel? Justify your answer.

a=(1,2,32),\quad b=(2,4,64)

The angle formula requires two nonzero vectors; a dot product is a scalar.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

b=\lambda a

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$(2,4,64)=2(1,2,32)$
All components have the same scalar multiplier.

The requested relation or conclusion is shown below.

b=2a

Checks and common pitfalls: All components have the same scalar multiplier.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

14 / Transfer#Your turn

Explain why these three vectors do not give unique coefficients.

e_1=(1,0,0), e_2=(0,1,0), e_3=(33,1,0)

A basis must be independent. Weights representing a point in a plane sum to one.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

e_3=te_1+e_2

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$(33,1,0)=33(1,0,0)+(0,1,0)$
The dependent triple permits more than one representation.

The requested relation or conclusion is shown below.

e_3=33e_1+e_2

Checks and common pitfalls: The dependent triple permits more than one representation.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

15 / Transfer#Your turn

A unit direction has cos²α=1/t², cos²β=1/9. Find cos²γ.

t=34

Projection requires a nonzero direction; signed scalar projection can be negative.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the dot product to connect coordinates with length or angle.

Hint 2

Use this intermediate relation.

cos²α+cos²β+cos²γ=1

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$\cos^2\gamma=1-1/1156-1/9$
The squared direction cosines sum to one.

The requested value is 0.888023836986.

Checks and common pitfalls: The squared direction cosines sum to one.

Think first. Reveal a hint when the class is ready.

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.