Vector addition — triangle law, parallelogram law: Question bank

Two anchors you will lean on throughout:

The angle in every formula, $θ$ , is the angle between the two vectors measured tail-to-tail — both arrows starting from one shared point.
The two master formulas: $R = A^{2} + B^{2} + 2 A B cos θ$ (magnitude) and $tan α = \frac{B sin θ}{A + B cos θ}$ (direction, angle from $A$ ).

True or false — justify

Adding two vectors of length 3 and 4 can never give a resultant of length 8.

True. The resultant is capped at

A + B = 7

(reached only when

θ = 0^{\circ}

), so

8

is impossible — it lies outside the allowed band

[∣ A - B ∣, A + B] = [1, 7]

Two vectors of length 3 and 4 can produce a resultant of length 1.

True. That is the minimum,

∣ A - B ∣ = 1

, reached when they point in exactly opposite directions (

θ = 18 0^{\circ}

A + B

has the same magnitude as

A - B

, then

A

and

B

are perpendicular.

True. Squaring both,

A^{2} + B^{2} + 2 A B cos θ = A^{2} + B^{2} - 2 A B cos θ

forces

cos θ = 0

, i.e.

θ = 9 0^{\circ}

(a fact you can also see with the Scalar (dot) product, since

A \cdot B = 0

The resultant of two vectors always lies between them (inside the angle they make).

True (for the angle they subtend tail-to-tail). The diagonal of the parallelogram sits between its two adjacent sides, so the resultant's direction is always squeezed between

A

and

B

For two equal-length vectors, the resultant bisects the angle between them.

True. Symmetry: swapping the two identical vectors leaves the figure unchanged, so the resultant cannot favour either — it must lie on the angle bisector.

The parallelogram law gives a different answer from the triangle law for the same two vectors.

False. They are the same operation in disguise; the parallelogram's opposite side equals

B

, so its diagonal closes exactly the head-to-tail triangle of

A

and

B

Vector addition is commutative:

A + B = B + A

True. Whether you walk journey

A

then

B

, or

B

then

A

, you land at the same corner of the parallelogram — same endpoint, same resultant.

You can add a

5 N

force to a

5 m/s

velocity if their magnitudes match.

False. Only like quantities add; a force and a velocity live in different unit-worlds, so their sum is meaningless no matter the numbers.

θ

increases from

0^{\circ}

18 0^{\circ}

, the resultant magnitude decreases the whole way.

True.

R^{2} = A^{2} + B^{2} + 2 A B cos θ

and

cos θ

falls monotonically from

+ 1

- 1

[0^{\circ}, 18 0^{\circ}]

, dragging

R

down with it.

Three vectors can add to the zero vector only if they are all equal in magnitude.

False. Any three vectors that form a closed head-to-tail triangle sum to zero — the sides need not be equal, just able to close.

Spot the error

3 N

and

4 N

act at

6 0^{\circ}

, so

R = 3 + 4 cos 6 0^{\circ} = 5 N

Wrong: they dropped the

A^{2} + B^{2}

structure entirely. Correct is

R = 9 + 16 + 2 (3) (4) (0.5) = 37 \approx 6.08 N

— you cannot add a projection to a magnitude.

"To find

A + B

by the triangle law, I draw both vectors from the same point and join their tips."

That is tail-to-tail joining of tips, which gives the difference

B - A

(see Subtraction of vectors and the difference vector). For the sum you place

B

's tail on

A

's head (head-to-tail); the closing side is

A + B

"In the parallelogram law the resultant is one of the sides of the parallelogram."

No — the sides are

A

and

B

. The resultant is the diagonal through the common tail, not a side.

θ

R = A^{2} + B^{2} + 2 A B cos θ

is the angle

B

makes with the horizontal."

θ

is the angle between the two vectors, measured tail-to-tail — nothing to do with a coordinate axis. Redraw both from a common origin and read the wedge between them.

"Since

tan α = \frac{B sin θ}{A + B cos θ}

, the resultant makes angle

α

with the horizontal."

α

is measured from

A

, not from the horizontal. Only if

A

itself lies along the horizontal do the two coincide.

R = A^{2} + B^{2} + 2 A B cos θ

is just Pythagoras, so it needs a right angle."

It reduces to Pythagoras only at

θ = 9 0^{\circ}

(where the

2 A B cos θ

term vanishes). In general it is the Law of cosines applied to the addition triangle, valid at any angle.

"For

θ = 18 0^{\circ}

R = A^{2} + B^{2} - 2 A B = A - B

The square root gives

(A - B)^{2} = ∣ A - B ∣

, an absolute value. If

B > A

the resultant still has positive length

B - A

and points along

B

Why questions

Why is the resultant always shorter than

A + B

unless the vectors are parallel?

Because the head-to-tail path bends at the join; a bent path from start to finish is always shorter than the two segments laid straight (triangle inequality). Only a

0^{\circ}

bend keeps the full length

A + B

Why does

cos θ

— and not

sin θ

— appear in the magnitude formula?

cos θ

measures how much of

B

points along

A

; that aligned part is what stretches the resultant. This is exactly the piece the Scalar (dot) product

A \cdot B = A B cos θ

isolates.

Why must

θ

be measured tail-to-tail rather than head-to-tail?

The formulas were derived from the parallelogram built with both tails at one point; the "opening" angle between the arrows as they leave that point is what controls the geometry. Head-to-tail measurement gives the supplement,

18 0^{\circ} - θ

, and flips the sign of the cross term.

Why can we resolve

B

into

(B cos θ, B sin θ)

and add components separately?

Because displacement along

x

and along

y

are independent journeys — the eastward part of a trip is unaffected by its northward part (see Resolution of a vector into components and Vectors — components and unit vectors).

Why does the direction formula use

R_{y} / R_{x}

instead of some average of the input angles?

The resultant's tilt is fixed by where its head lands, whose vertical rise

R_{y} = B sin θ

over horizontal run

R_{x} = A + B cos θ

defines

tan α

. Averaging the input directions ignores the vectors' lengths, so it fails when

A \neq = B

Why is relative velocity (boat crossing a river) a vector-addition problem, not a subtraction one?

The velocity of the boat relative to ground is the boat-in-water velocity plus the water-relative-to-ground velocity — two journeys chained head-to-tail. See Relative velocity for how the current shifts the resultant heading.

Edge cases

What is the resultant when one of the two vectors is the zero vector?

The sum is the other vector unchanged:

A + 0 = A

. The zero vector is a journey of no length, so it adds nothing and has no direction to shift the result.

What happens to the direction formula when

A + B cos θ = 0

The denominator vanishes,

tan α \to \infty

, meaning

α = 9 0^{\circ}

: the resultant is exactly perpendicular to

A

. This occurs when

B

's backward pull cancels all of

A

's length along the base.

Two equal vectors point in exactly opposite directions (

θ = 18 0^{\circ}

A = B

). What is the resultant?

The zero vector.

R = ∣ A - B ∣ = 0

; they are a perfectly cancelling pair, and the direction is undefined because there is no leftover journey.

θ \to 0^{\circ}

, what does the parallelogram degenerate into?

It collapses to a line segment — the two vectors lie on top of each other, the "parallelogram" has zero width, and its diagonal becomes the straight sum

A + B

pointing the same way.

Can the resultant of two nonzero vectors point opposite to both of them?

No. The resultant always lies within the wedge between the two vectors, so it can never point outside that region — much less opposite to both.

If both vectors have the same magnitude and

θ = 12 0^{\circ}

, what is the resultant length?

R = A^{2} + A^{2} + 2 A^{2} cos 12 0^{\circ} = 2 A^{2} - A^{2} = A

— the resultant equals a single vector's length, and by symmetry bisects the

12 0^{\circ}

angle at

6 0^{\circ}

from each.

Recall One-line survival kit

Angle is tail-to-tail · $R \in [∣ A - B ∣, A + B]$ · $R = A^{2} + B^{2} + 2 A B cos θ$ · head-to-tail = sum, tips-joined = difference · diagonal = resultant, sides = the vectors.

Connections

Vector addition — triangle law, parallelogram law (parent)
Subtraction of vectors and the difference vector
Resolution of a vector into components
Vectors — components and unit vectors
Relative velocity
Law of cosines
Scalar (dot) product