Explanation:
A vector is defined by its magnitude and direction together.
When vectors are added, both their sizes and directions are combined.
So vector addition always takes magnitude and direction into account.

Explanation:
The components are mutually perpendicular.
Magnitude of the original vector is found by
R = √(5² + 12²).
R = √(25 + 144)
= √169
= 13 units.
So the components combine by the Pythagorean relation to give the original vector.

Explanation:
The components are at right angles, so
R = √(6² + 8²).
R = √(36 + 64).
R = √100 = 10 N.
So the components recombine to give the original 10 N.

Explanation:
Zero resultant requires that the final head comes back to the initial tail.
When three vectors are arranged head-to-tail and close to form a triangle, this condition is met.
Thus three vectors giving zero sum are represented as a closed triangle.

Explanation:
Each vector is first split into its x- and y-components.
All x-components are combined algebraically to get the resultant x-component.
Similarly all y-components are combined to get the resultant y-component.
These two components together define the resultant vector.

Explanation:
In the parallelogram law, A and B are drawn from the same point as adjacent sides.
A parallelogram is completed on these two vectors.
The diagonal from the common tail gives the vector sum A + B.

Explanation:
When vectors are arranged head-to-tail and form a closed triangle, start and end points coincide.
This means there is no overall change in position.
So the resultant or net displacement is zero.

Explanation:
Polygon law is an extension of the triangle law.
Vectors are placed head-to-tail in sequence to form a polygon.
The side joining the first tail and the last head gives the resultant.
This law allows addition of any number of vectors together.

Explanation:
Resultant magnitude must lie between |10 – 6| and 10 + 6.
So it must be between 4 and 16 inclusive.
Values 4, 10 and 16 fall in this range and are possible.
A magnitude of 2 units is less than 4, so it is not possible.

Explanation:
Commutative property means A + B = B + A, which holds for vectors.
Associative property means
(A + B) + C = A + (B + C),
which also holds.
Therefore vector addition is both commutative and associative.

Explanation:
The components are 4 km east and 3 km north.
Resultant lies between north and east.
Larger component is along east, so direction is closer to east.
This is described as a direction east of north.

Explanation:
Use the identities
|A ± B|² = A² + B² ± 2AB cosθ.
Given |A + B|² = |A – B|²,
the 2AB cosθ terms must cancel.
That is possible only if cosθ = 0.
So the angle between A and B is 90°.

Explanation:
The two vectors are equal and perpendicular.
For perpendicular vectors,
R = √(A² + B²).
Here R = √(5² + 5²).
R = √(25 + 25)
= √50.
So the magnitude of the resultant is 5√2.

Explanation:
Take east as positive direction and west as negative.
Net displacement = 6 – 8 = –2 km.
Negative sign shows direction is towards west.
Magnitude of displacement is 2 km.

Explanation:
Zero resultant means net effect of the two vectors is nothing.
Graphically, one vector must exactly cancel the other.
This appears as two equal arrows on the same line pointing in opposite directions.
So they overlap on a straight line segment with opposite arrows.

Explanation:
Minimum resultant occurs when vectors act in opposite directions.
Opposite directions correspond to an angle of 180° between them.
So for magnitudes 7 and 24, the least resultant is obtained at 180°.

Explanation:
Resultant is smallest when the vectors point in opposite directions.
Then the effective magnitude is the difference of their magnitudes.
Thus the minimum possible resultant is ||A| – |B||, written as |A – B|.

Explanation:
Resultant is largest when the vectors point in the same direction.
In that case their magnitudes simply add.
So maximum R is |A| + |B|,
which we write as |A + B|.

Explanation:
First vector is drawn from a point, and the second is drawn from the head of the first.
These two sides in order represent the two vectors being added.
The side joining the tail of the first to the head of the second gives the sum.
Thus the third side of the triangle taken in opposite order represents the resultant.

Explanation:
Two vectors are drawn from a common point as adjacent sides of a parallelogram.
A parallelogram is then completed on these two sides.
The diagonal passing through the common tail gives the sum of the vectors.
This construction is known as the parallelogram law of vector addition.

Explanation:
Forces towards north and east are at right angles.
So we use the Pythagorean relation for perpendicular vectors.
R = √(5² + 12²).
R = √(25 + 144) = √169.
Thus the resultant force has magnitude 13 N.

Explanation:
A + B = 0 means that the two vectors cancel each other exactly.
For this to happen, one must be equal in length but opposite in direction to the other.
Geometrically they are collinear but point in opposite directions.
So they are antiparallel with equal magnitudes.

Explanation:
Opposite collinear vectors subtract in magnitude.
Take the larger force direction as positive.
Net magnitude is |10 – 6|.
So the resultant force has magnitude 4 N.

Explanation:
Collinear vectors in the same direction simply add in magnitude.
The total effect is obtained by adding their lengths.
So the magnitude of the resultant is |A| + |B|.
This is written as A + B when we talk about magnitudes only.

Explanation:
In the head-to-tail method, the tail of B is placed at the head of A.
The free tail is at the start of A and the free head is at the end of B.
The resultant vector is drawn from the free tail to the free head.
So it starts from tail of A and ends at head of B.

Explanation:
Let each vector have magnitude A and angle between them be θ.
Resultant magnitude is R = √(A² + A² + 2A² cosθ).
Condition is R = A,
so A² = 2A²(1 + cosθ).
Dividing by A² gives
1 = 2(1 + cosθ),
so 1/2 = 1 + cosθ.
Thus cosθ = –1/2
and θ = 120°.

Explanation:
For any two vectors A and B with angle θ, resultant is R = √(A² + B² + 2AB cosθ).
Here A = B = 5 units and θ = 60°.
So R² = 25 + 25 + 2(5)(5) cos60°.
cos60° = 1/2,
so R² = 50 + 50/2
= 50 + 25 = 75.
Therefore R = √75
= 5√3 units.

Explanation:
The two forces are mutually perpendicular.
For such vectors, resultant R = √(A² + B²).
Here R = √(6² + 8²).
R = √(36 + 64) = √100.
So the magnitude of the resultant is 10 N.

Explanation:
The two displacements are at right angles to each other.
For perpendicular vectors we use Pythagoras to find the resultant.
R = √(4² + 3²).
R = √(16 + 9) = √25.
So the magnitude of the resultant displacement is 5 m.

Explanation:

• Scalar multiples of A always lie along the same line as A, so kA is parallel to A.

Explanation:

|3A| = |3|·|A| = 3·|A|

Explanation:

• If k=0, kA = 0 (the zero vector).
• If k>0, kA points in the same direction as A.
• If k<0, kA points opposite to A (–kA).

Explanation:

Collinear vectors lie along the same line, so they point in exactly the same (0°) or exactly opposite (180°) direction

Explanation:

|A × B| = A B sin θ, with θ the angle between A and B.

Explanation:

A “zero vector” has zero length and its direction is undefined; it is also called the “null vector.”

Explanation:

|A × B| = AB sin θ, and sin 0° = 0, so the result is the zero vector.

Explanation:

A · B = AB cos θ, and cos 90° = 0, so the result is 0.

Explanation:

A unit vector has been scaled so its length is exactly 1.

Explanation:

A + 0 = A by definition of the zero vector.

Explanation:

By Pythagoras’ theorem (since the angle is 90°), |R| = √(A² + B²).

Explanation:

“Commutative” means you can swap the order of the vectors without changing the resultant: A + B = B + A.

Explanation:

Graphical vector addition is done by placing the tail of one vector at the head of the other (head-to-tail). This is neither a scalar “simple addition” nor a product operation.

Explanation:

F₁ at 0° has components (10, 0),
F₂ at 90° has (0, 10).
Resultant R = (10 + 0, 0 + 10)
= (10, 10),
|R| = √(10² + 10²)
= √200
≈ 14.14 N 

Explanation:

Unit vector along 4î + 3ĵ , is obtained by dividing the present vector by its magnitude.

The magnitude of the given vector is 5. Hence, the required unit vector is (4î + 3ĵ)/5.

Explanation:

Consider the graphical representation of these two vectors. When one vector is added to the other in the same direction, the lengths will be added.

The resultant vector will bear the resultant length. Length is the magnitude of the vector. Hence the magnitudes add to give the magnitude of the resultant vector.

Explanation:

For three forces to have zero resultant, their magnitudes must satisfy the triangle inequality.

(each pair’s sum ≥ the third).

Here 10 + 20 = 30 < 40, so they can’t form a closed triangle and cannot balance to zero.

Explanation:

A – B = (î – 6ĵ) – (4î + 2ĵ)
= î – 6ĵ – 4î – 2ĵ
= (1 – 4)î + (–6 – 2)ĵ
= –3î – 8ĵ

|u| = |v| = 1 (unit vectors)
Angle between u and v = 90° (perpendicular)

Use the formula:
|u − v|² = |u|² + |v|² − 2|u||v|cosθ

= 1² + 1² − 2(1)(1)cos(90°)
= 1 + 1 − 0
= 2

Therefore,
|u − v| = √2

R = √(300² + 400²)
= 500 km.

R = √(V² + V² + 2·V·V·cos 60°)
= √(2V² + 2V²·½)
= √(2V² + V²)
= √(3V²)
= V√3

|A| = √(7² + 24²)
= √(49 + 576)
= √625 = 25.

We know:
cos(120°) = -1/2

So,
R = √[2P² + 2P²·(-1/2)]
R = √[2P² – P²]
R = √[P²]
R = P

Let the two unit vectors be A and B.
So,
|A| = |B| = 1
|A + B| = 1

Use the formula:
|A + B|² = |A|² + |B|² + 2|A||B|cosθ

⇒ 1² = 1² + 1² + 2(1)(1)cosθ
⇒ 1 = 1 + 1 + 2cosθ
⇒ 1 = 2 + 2cosθ
⇒ 2cosθ = -1
⇒ cosθ = -1/2
⇒ θ = 120°

Now, find |A − B|:
|A − B|² = |A|² + |B|² − 2|A||B|cosθ
= 1 + 1 − 2(1)(1)(cos120°)
= 2 − 2(−1/2)
= 2 + 1 = 3
⇒ |A − B| = √3

For resultant we use below formula: 

(2Q + 2P) + (2Q – 2P) = 4Q,
which is parallel to Q, so the angle between them is 0°.

B – A = 2ĵ
⇒ B = A + 2ĵ
= 2î + (3+2)ĵ + 2k̂
= 2î + 5ĵ + 2k̂
⇒ |B| = √(4+25+4)
= √33.

Plot A, then attach B head-to-tail; you return to the origin.

A and B are equal in magnitude but opposite in direction.

Their head-to-tail connection completes the path back to the start, you will get a straight line.

“Commutative” means order doesn’t matter.

Graphically, placing A then B, or B then A, yields the same head-to-tail resultant.

Magnitude and direction are identical either way.

This holds for all vectors in a vector space.

Steps:
(1) resolve into components,
(2) add all x’s and all y’s,
(3) recombine into R.

Parallelogram law is an equivalent graphical check.

At no point do you multiply components.

Option (d) is irrelevant to vector addition.

Pure x-axis resultant means no net vertical component.

Component addition: Rᵧ = Aᵧ + Bᵧ must be zero.

Horizontal sum Aₓ + Bₓ can be nonzero.

Hence only condition (a) guarantees R is along x.

Sum x-components: 8 + (–8) = 0
→ no horizontal part.

Sum y-components: 6 + 6 = 12
→ positive vertical part.

Zero x and positive y
⇒ a vector straight up along +y.

Parallelogram law uses the two vectors as adjacent sides.

The diagonal from the common origin is the resultant.

It decomposes into perpendicular components on x and y.

This visual proof also shows commutativity of addition.

Rₓ = 4 + (–2)
Rₓ = 2,
R_y = 3 + 6
R_y = 9.

θ = tan⁻¹(R_y/R_x)
= tan⁻¹(9/2).

Compute components first, then use tan⁻¹.

This yields the correct direction of R.

Sum x-components:
3 + (–5)
= –2
→ –2î.

Sum y-components:
–4 + 2
= –2
→ –2ĵ.

R = –2î + (–2ĵ) by component-wise addition.

Vector closure ensures the result is itself a vector.

Zero x-sum means no horizontal part; nonzero y-sum gives a pure vertical resultant.

If each component= V,
then R = Vî + Vĵ
use Pythagoras’ theorem
⇒ |R|
= √(V² + V²)
= V√2.

R = (5+0)î + (0+12)ĵ
= 5î + 12ĵ.
Magnitude = √(5² + 12²) = 13.

Subtract component-wise:
(î+ĵ) – (2î+7ĵ)
= (1–2)î + (1–7)ĵ
X-component:
1 – 2
= –1
→ –î
Y-component:
1 – 7 = –6 → –6ĵ
Resultant: –î –6ĵ

Add x-components:
2 + 1 = 3
Add y-components:
7 + 1 = 8
Result: 3î + 8ĵ

On adding two vectors we get a vector.
Vector addition preserves both magnitude and direction.
Use the head-to-tail or parallelogram law to find the resultant.
Vectors are closed under addition, so the result is always another vector.