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).

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

Explanation:

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

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:

Start with the equation:
|A + B| = |A - B|

Square both sides:
A² + B² + 2AB cos(θ)= A² + B² - 2AB cos(θ)

Cancel A² and B² from both sides:
2AB cos(θ) = -2AB cos(θ)

Move everything to one side:
4AB cos(θ) = 0

Since A and B are not zero, cos(θ) must be zero: cos(θ) = 0 → θ = 90°

Explanation:

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

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:
The components are 4 km east and 3 km north.
Resultant lies between north and east.

Because east (4 km) is larger than north (3 km), the resultant is closer to east.
So we describe it as north of east (start from east, tilt toward north).

“East of north” would mean it’s closer to north (north component larger), which is not the case here.

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

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:
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:
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.

R = √(300² + 400²)
= 500 km.

Explanation:

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

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.

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

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.

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

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

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:

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:

Force is a vector because it has magnitude and direction.
Kinetic energy is a scalar because it has magnitude only and no direction.

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:
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.

For resultant we use below formula: 

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:
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:
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:

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

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.

We know:
cos(120°) = -1/2

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

“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.

|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

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:
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:

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

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

Explanation:

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

Explanation:

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

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:
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:
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.

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.

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:

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.

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.

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.

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

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.

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.

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

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.

Explanation:

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

Explanation:

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

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

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.

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:
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.

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.

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.

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:

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

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:
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:

|3A| = |3|·|A| = 3·|A|

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:

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

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:

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

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.

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.

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:

Rectangular form of any vector uses its x-component with î and y-component with ĵ.
Aₓ and Aᵧ are the correct symbols for components along x and y axes.
So A = Aₓ î + Aᵧ ĵ is the standard and correct representation.

Explanation:

From A + B = A − B,
subtract A from both sides
→ B = −B.
So 2B = 0
→ B = 0 vector (only way this can happen).
A can be any vector; only B must be zero.