Explanation:

Vector 2B
= 2 × (1 i – 2 j + 2 k)
= (2 i – 4 j + 4 k)

Resultant R
= A – 2B
  = (3 i + 4 j – 5 k) – (2 i – 4 j + 4 k)
  = (3 – 2) i + (4 – (-4)) j + (-5 – 4) k
  = 1 i + 8 j – 9 k

|R| = √(1² + 8² + (-9)²)
  = √(1 + 64 + 81)
  = √146

Explanation:

Add components:
Rₓ = 4 + (−1) = 3;
Rᵧ = −2 + 5 = 3.
Thus
tanθ = Rᵧ/Rₓ
= 3/3 = 1,
giving θ = 45°.

Explanation:

Since the vectors are at right angles, square the magnitudes and sum:
R² = 2² + 3² + 6²
= 4 + 9 + 36
= 49, hence
R = √49 = 7.

Explanation:

By the parallelogram law,
R² = 5² + 7² + 2·5·7·cos 60°.
Since cos 60° = 0.5,
R² = 25 + 49 + 35
= 109,
so R = √109.

Explanation:

First compute the components:
Rₓ = 3 + 2 = 5, Rᵧ
= 4 + (−1) = 3.
The magnitude is
√(Rₓ² + Rᵧ²)
= √(5² + 3²)
= √(25 + 9)
= √34.

Explanation:

Use head to tail rule:

Explanation:

Explanation:

What displacement must be added to the displacement
25 î − 6 ĵ m to give a displacement of 7.0 m pointing in the x-direction?

Solution:

Let the required displacement vector be:
B = x î + y ĵ

We are given:
A = 25 î − 6 ĵ
Resultant R = 7 î + 0 ĵ

We know:
A + B = R

So:
(25 + x) î + (−6 + y) ĵ = 7 î + 0 ĵ

Now equate components:

i-component:
25 + x = 7 → x = 7 − 25 = −18

j-component:
−6 + y = 0 → y = 6

So the required displacement is:
B = −18 î + 6 ĵ

Explanation:

Explanation:

Explanation:

Explanation:

To find the resultant of two forces, we use vector addition. The resultant depends on the angle between them.

Maximum resultant (when both forces act in the same direction, angle = 0°):
R = 6 + 8 = 14 N

Minimum resultant (when both forces act in opposite directions, angle = 180°):
R = |8 – 6| = 2 N

Therefore, the resultant of 6 N and 8 N can be any value between 2 N and 14 N depending on the angle.

Explanation:

Largest ≤ sum of other two

If three vectors add to zero, they must form a triangle. According to the triangle inequality, the sum of the magnitudes of any two vectors must be greater than or equal to the third.

Let’s check each option:

Option (1): 2, 4, 8
→ 2 + 4 = 6 < 8 → ❌ Cannot form triangle
Invalid

Option (2): 4, 8, 16
→ 4 + 8 = 12 < 16 → ❌ Cannot form triangle
Invalid

Option (3): 1, 2, 1
→ 1 + 2 = 3 > 1 ✅
→ 2 + 1 = 3 > 1 ✅
→ 1 + 1 = 2 = 2 ✅
→ This is a degenerate triangle (straight line) → Resultant = 0

Explanation:

Explanation:

Orthogonality ⇒ A·B = 0

A·B = cos t · cos (t⁄2) + sin t · sin (t⁄2)
= cos [t − t⁄2]
= cos (t⁄2).

Set cos (t⁄2) = 0
⇒ t⁄2 = π⁄2 + nπ
⇒ t = π + 2nπ.
Smallest positive solution: t = π.

Explanation:

Explanation:

Video Explanation:

Explanation:

Explanation:

The angle between two vectors can never be greater than 180^o.

So, on increasing the θ, the magnitude of resultant vectors decreases.

Explanation:

The sum of the vectors cannot be equal to the sum of their unit vectors. 

Detailed Explanation:

P = Q implies both vectors have the same magnitude and direction.

Therefore P̂ = Q̂ (unit vectors match) and |P| = |Q|, so options A and B are true.

In option C, since Q̂ = P̂, both sides reduce to P P̂, so it is also true.

Option 4 equates 2P (because P + Q = 2P) with 2 P̂ (sum of two unit vectors). Unless |P| = 1, those vectors are not equal, so statement D is generally false.

Explanation:

Explanation:

The resultant of these two forces would lie in the north-east direction. To balance this resultant, the third force should be in a south-west direction.

Explanation:

Explanation:

Let A and B be the vectors.
Given |A + B| = |A − B|.

Square both sides:

|A + B|² = |A − B|²
⇒ (A + B)·(A + B) = (A − B)·(A − B)
⇒ |A|² + |B|² + 2 A·B = |A|² + |B|² − 2 A·B

Cancel the common terms |A|² + |B|²:

2 A·B = −2 A·B
⇒ 4 A·B = 0
⇒ A·B = 0

A zero dot product means the vectors are perpendicular, hence the angle between them is 90°.

Explanation:

For 17 N both the vector should be parallel i.e., the angle between them should be zero.

For 7 N both the vectors should be antiparallel i.e., the angle between them should be
180°.

For 13 N both the vectors should be perpendicular to each other i.e., the angle between them should be
90°.

Explanation:

Hint: Use Resultant of vectors.
Step: Find the relation between v1 and v2

Explanation:

Original resultant: R1 = A + B
New resultant after doubling B: R2 = A + 2B

Given that R2 is perpendicular to A:
A · R2 = 0
⟹ A · (A + 2B) = 0

Expand the dot product:
A·A + 2 A·B = 0
⟹ A² + 2ABcosθ = 0
⟹ ABcosθ = −A² / 2 … (✱)

Find |R1|²:
|R1|² = (A + B) · (A + B)
= A² + B² + 2ABcosθ

Substitute (✱):
|R1|² = A² + B² − A² = B²

Therefore |R1| = |B|, so R1 equals B in magnitude and direction.

Explanation:
Draw the three equal forces at 120° to each other head to tail.
They form a closed equilateral triangle or hexagon-like figure when arranged properly.
A closed polygon of vectors indicates that their vector sum is zero.
So the resultant of the three equal forces is zero.

Explanation:
For two equal vectors, the resultant lies along the bisector of the angle between them.
If resultant is at 30° to one force, it is also at 30° to the other force.
So the total angle between the two forces is 30° + 30° = 60°.

Explanation:
Boat velocity and current velocity are at right angles to each other.
Take east as x-direction and north as y-direction.
Resultant speed R = √(15² + 8²).
R = √(225 + 64).
R = √289 = 17 m/s.

Explanation:
When vectors act in the same direction, their magnitudes add directly.
So resultant magnitude is 4 units + 9 units.
That gives a resultant of 13 units.

Explanation:
Two non-zero vectors can add to give zero only when they cancel each other.
Cancellation requires equal magnitudes and opposite directions.
In that special case the resultant is zero even though each vector is non-zero.

Explanation:
Take east as x-direction and north as y-direction.
The two displacements are at right angles.
Resultant displacement R = √(5² + 12²).
R = √(25 + 144).
R = √169 = 13 km.

Explanation:
Let the vectors be A and B with angle θ between them.
Given |A + B| = |A − B| so their squares are also equal.
Using |A ± B|² = A² + B² ± 2AB cosθ
gives A² + B² + 2AB cosθ
= A² + B² − 2AB cosθ.
This simplifies to 4AB cosθ = 0 so cosθ = 0.
Therefore the angle between the vectors is 90°.

Explanation:
The forces are equal and perpendicular.
Using Pythagoras, R = √(20² + 20²).
R = √(400 + 400).
R = √800 = 20√2 N.

Explanation:
Maximum resultant occurs for forces in the same direction and equals P + Q.
Minimum resultant occurs for forces in opposite directions and equals |P − Q|.
So the resultant must always lie between |P − Q| and P + Q.

Explanation:
Use R = √(A² + B²) for perpendicular forces.
R = √(7² + 24²).
R = √(49 + 576).
R = √625 = 25 N.

Explanation:
Zero resultant means the forces cancel each other exactly.
For that, they must have equal magnitudes and opposite directions.
Equal magnitudes give P = Q and opposite directions correspond to θ = 180°.

Explanation:
The components are at right angles to each other.
Resultant of perpendicular components is found using R = √(6² + 8²).
R = √(36 + 64).
R = √100 = 10 N.
So the resultant equals the original 10 N force.

Explanation:
When forces are represented by sides of a triangle taken in order, the head of the last meets the tail of the first.
This means the path closes back on itself.
A closed polygon for force vectors indicates that their vector sum is zero.
So the resultant of the three forces is zero.

Explanation:
Let each force have magnitude F and angle between them be 60°.
Using R² = F² + F² + 2F² cos60°
gives R² = 2F² + 2F²(1/2).
This simplifies to R² = 3F².
Given R = 10 N,
so 10² = 3F².
Thus 100 = 3F²
and F² = 100 / 3,
so F = 10 / √3

Explanation:
For two forces represented as adjacent sides of a parallelogram, the resultant is shown by the diagonal.
This diagonal is drawn from the common tail of the two forces.
Therefore the direction of the resultant is along that diagonal.

Explanation:
Let the forces be P and Q with P > Q and angle between them be θ.
Given that resultant equals the larger force P.
Condition R = P means the smaller force Q must cancel part of P.
This is possible only when they act in opposite directions, so θ = 180°.

Explanation:
The forces are perpendicular, so use R = √(A² + B²).
R = √(5² + 12²).
R = √(25 + 144).
R = √169 = 13 N.

Explanation:
Let each force have magnitude F and angle between them be 120°.
Using R² = F² + F² + 2F² cos120°
gives R² = 2F² + 2F²(−1/2).
That simplifies to
R² = 2F² − F² = F².
So R = F, which equals the magnitude of either force.

Explanation:
When two vectors act at any angle θ, we use the law of cosines.
This gives R² = P² + Q² + 2PQ cosθ.
Taking square root on both sides gives
R = √(P² + Q² + 2PQ cosθ).

Explanation:
Minimum resultant occurs when one force tends to cancel the other.
This happens when they act in exactly opposite directions.
Opposite directions correspond to an angle of 180° between P and Q.

Explanation:
Resultant depends on both magnitudes and angle between the forces.
The cosine formula shows that R is largest when cosθ is maximum.
cosθ is maximum at θ = 0°, meaning both forces act in the same direction.

Explanation:
Minimum resultant occurs when the forces act in opposite directions.
In that case the effective magnitude is the difference of their magnitudes.
So minimum resultant is |8 N − 6 N| = 2 N.

Explanation:
Maximum resultant occurs when forces act in the same direction.
In that situation their magnitudes simply add.
So maximum resultant is 6 N + 8 N = 14 N.

Explanation:
For two vectors at 90° the law of cosines reduces to Pythagoras.
That gives R² = P² + Q².
So the correct relation for the resultant is R² = P² + Q².

Explanation:
The forces are equal and perpendicular to each other.
For perpendicular vectors, resultant is R = √(P² + P²).
R = √(2P²).
R = P√2.

Explanation:
Both forces have magnitude 10 N and the angle between them is 60°.
Use R = √(A² + B² + 2AB cosθ).
R² = 10² + 10² + 2(10)(10) cos60°.
cos60° = 1/2, so R² = 100 + 100 + 200(1/2).
R² = 200 + 100 = 300, so R = √300 = 10√3 N.

Explanation:
When two equal vectors are in opposite directions, their effects cancel.
Mathematically they have the same magnitude but opposite directions.
Adding them gives a net vector of zero magnitude.
So the resultant is zero.

Explanation:
Use the formula R = √(A² + B² + 2AB cosθ) for any two vectors.
Here A = B = 5 N and θ = 120°.
cos120° is equal to −1/2.
So R² = 25 + 25 + 2(5)(5)(−1/2).
R² = 50 − 25
= 25, so R = 5 N.

Explanation:
The forces are mutually perpendicular.
Resultant of perpendicular vectors is given by R = √(A² + B²).
Here R = √(6² + 8²).
R = √(36 + 64).
R = √100 = 10 N.

Explanation:
The two forces are at right angles along east and north.
For perpendicular vectors, the resultant magnitude is found by Pythagoras.
R = √(3² + 4²).
R = √(9 + 16).
R = √25 = 5 N.