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Vectors · eight-question check

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

TOPIC 01

Vectors

Try two tasks per skill independently, then revisit the linked skill.

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

Which vector equals (2,−1), regardless of its drawn start?

Skills and prerequisite lessons

Scope: JM02

Option A $(2,-1)$
Option B $(-2,1)$
Option C $(1,-2)$
Option D $(2,1)$

Working and explanation

BUILD THE REASONING

Hint 1

Equality preserves both components.

Hint 2

A translation changes neither component.

Worked solution

Compare horizontal and vertical displacements.
$(2,-1)=(2,-1)$
Only A preserves both signs and lengths.

Checks and common pitfalls: The position of a free arrow is irrelevant; orientation is not.

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

02 / Foundation#Your turn

If |u|=4, what is |−2u|?

Skills and prerequisite lessons

Scope: JM02

Option A $-8$
Option B $2$
Option C $8$
Option D $6$

Working and explanation

BUILD THE REASONING

Hint 1

Separate the sign from the scale.

Hint 2

Use the absolute multiplier.

Worked solution

Multiply lengths by two.
$|-2u|=2|u|$
Apply the given norm.
$2\cdot4=8$

C: 8.

Checks and common pitfalls: The minus sign reverses the arrow only.

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

03 / Foundation#Your turn

For A=(2,−1), B=(−1,3), choose AB.

Skills and prerequisite lessons

Scope: JM02

Option A $(-3,4)$
Option B $(3,-4)$
Option C $(1,2)$
Option D $(-3,2)$

Working and explanation

BUILD THE REASONING

Hint 1

Use B−A.

Hint 2

Subtract −1 carefully.

Worked solution

Horizontal displacement is −1−2.
$AB_x=-3$
Vertical displacement is 3−(−1).
$AB_y=4$

A: (−3,4).

Checks and common pitfalls: Both coordinates follow the same end-minus-start order.

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

04 / Foundation#Your turn

A=(0,0), B=(9,0), AP:PB=2:1 internally. Choose Pₓ.

Skills and prerequisite lessons

Scope: JM02

Option A $3$
Option B $6$
Option C $4.5$
Option D $18$

Working and explanation

BUILD THE REASONING

Hint 1

There are three equal parts.

Hint 2

AP occupies two of them.

Worked solution

One part is length three.
$9/(2+1)=3$
Take two parts from A.
$P_x=2\cdot3=6$

B: 6.

Checks and common pitfalls: The ratio compares two pieces, not a piece with the whole.

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

05 / Foundation#Your turn

Choose (1,2)·(3,4).

Skills and prerequisite lessons

Scope: JM02

Option A $(3,8)$
Option B $11$
Option C $21$
Option D $-5$

Working and explanation

BUILD THE REASONING

Hint 1

Dot product is scalar.

Hint 2

Add the corresponding products.

Worked solution

Compute the component products.
$1\cdot3=3,\quad 2\cdot4=8$
Add them.
$3+8=11$

B: 11.

Checks and common pitfalls: Dot product and determinant answer different questions.

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

06 / Foundation#Your turn

Project v=(3,4) onto u=(2,0): choose the coefficient t in tu.

Skills and prerequisite lessons

Scope: JM02

Option A $3$
Option B $5$
Option C $1.5$
Option D $6$

Working and explanation

BUILD THE REASONING

Hint 1

The projected vector is (3,0).

Hint 2

Solve 2t=3.

Worked solution

Match the horizontal component.
$(2t,0)=(3,0)$
Divide by two.
$t=3/2$

C: 1.5.

Checks and common pitfalls: The scalar projection and the coefficient agree only for a unit direction.

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

07 / Foundation#Your turn

D,E are midpoints of AB,AC. Choose directed DE.

AB=u,\ AC=v

Skills and prerequisite lessons

Scope: JM02

Option A $(v-u)/2$
Option B $(u+v)/2$
Option C $v-u$
Option D $(u-v)/2$

Working and explanation

BUILD THE REASONING

Hint 1

D has position u/2.

Hint 2

E−D gives the required direction.

Worked solution

Subtract midpoint positions.
$DE=v/2-u/2$
Factor one half.
$DE=(v-u)/2$

Checks and common pitfalls: A diagram does not determine a directed sign without an endpoint order.

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

08 / Foundation#Your turn

Sides 3 and 4 enclose a right angle. Choose the opposite side length.

Skills and prerequisite lessons

Scope: JM01 / JM02

Option A $7$
Option B $1$
Option C $5$
Option D $25$

Working and explanation

BUILD THE REASONING

Hint 1

The dot product vanishes.

Hint 2

Use squared lengths first.

Worked solution

Use the cosine rule with cosine zero.
$a^2=9+16=25$
Take the positive square root.
$a=5$

C: 5.

Checks and common pitfalls: The law concerns squares before converting back to a side length.

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