Standard unit vectors follow i × j = k.
Here A × B = (2i) × (3j)
= 6 (i × j).
So A × B = 6k.

Explanation:

Explanation:

For parallel vectors, the angle between them is 0°.
Sine of 0° is zero.
So |A × B| = |A||B| sin0° = 0, meaning the cross product is the zero vector.

Explanation:

B is 2·A (parallel to A), so A × B = 0 for any parallel vectors.

Cross product is defined to give a new vector quantity.
This new vector is perpendicular to the plane containing the original two vectors.
So the result of A × B is always a vector, not a scalar.

For vectors A and B with angle θ between them, the cross product magnitude is given by |A × B|.
By definition, |A × B| = |A||B| sinθ.
So the magnitude depends on both magnitudes and sine of the included angle.

Explanation:

Explanation:

Cross product is zero if vectors are parallel (θ = 0° or 180°) or if either vector is a zero vector.
Because sin0° = sin180° = 0.

Vector A and λA have the same or opposite direction depending on sign of λ.
So they are parallel or antiparallel vectors.
Cross product of parallel vectors is zero, hence A × (λA) is zero vector.

Cross product magnitude is |A × B| = |A||B| sinθ.
If A and B are non-zero and A × B = 0, then sinθ must be zero.
Sine becomes zero at θ = 0° or 180°, meaning the vectors are parallel or antiparallel.

Cross product produces a vector that is normal to the plane of the two vectors.
This new vector is at right angle to both A and B.
So the direction of A × B is perpendicular to the plane containing A and B.

Magnitude of cross product is |A × B| = |A||B| sinθ.
This is maximum when sinθ is maximum.
Sine has maximum value 1 at θ = 90°, so cross product is maximum at 90°.

Torque magnitude is |τ| = |r × F|.
By cross product definition, |r × F| = r F sinθ.
Here θ is the angle between r and F.

Explanation:

|A × B| = |A| |B| sin θ
A · B = |A| |B| cos θ

Given:
|A| |B| sin θ = √3 (|A| |B| cos θ)
⇒ sin θ = √3 cos θ
⇒ tan θ = √3
⇒ θ = 60°

Explanation:

The vector (B × A) is perpendicular to both B and A.
So when we take the **dot product** of (B × A) with A, the result is:

(B × A) · A = 0

Reason: dot product of two perpendicular vectors is zero. 

We know that i × j = k in right-handed system.
Reversing the order gives j × i = −(i × j).
So j × i = −k.

From cross product magnitude, |A × B| = |A||B| sinθ.
Given |A × B| equals |A||B|, so sinθ must be 1.
Sine becomes 1 at θ = 90°, so the vectors are perpendicular.

The vector r × F represents torque or moment of force.
Force has unit newton and position vector has unit metre.
So unit of torque is newton metre (N m).

Associative law would require (A × B) × C to equal A × (B × C).
For cross product these two expressions usually give different vectors.
Therefore vector product does not satisfy associative law.

Magnitude |A × B| = |A||B| sinθ depends on sine of angle between them.
It is maximum when sinθ is maximum.
Sine function reaches its maximum value 1 at 90°, so perpendicular vectors give largest |A × B|.

Magnitude |A × B| gives area of parallelogram formed by A and B.
A triangle formed by the same two vectors has half of this area.
So ½|A × B| represents area of the triangle formed by A and B.

Explanation:

|A × B| = |A| |B| sin θ
= 5 × 7 × sin 90°
= 35.

Explanation:

A × B = –(B × A)
So, the two vectors are equal in magnitude but opposite in direction,
hence the angle between them is π radians (180°).

Magnitude of vector product depends on sine of angle between vectors.
By definition, |A × B| = |A||B| sinθ.
So the cross product magnitude is proportional to sine of the included angle.

Explanation:

Explanation:

Calculate cross product using determinant:

Magnitude |A × B| equals |A||B| sinθ.
This is equal to the area of a parallelogram having sides A and B.
So cross product magnitude represents area of the parallelogram formed by two vectors.

Explanation:

A · B = 3×6 + 4×8
= 18 + 32 = 50 ❌ (not 48)

Vector product in components is written using a 3 × 3 determinant.
First row contains unit vectors i, j, k.
Second and third rows contain components of A and B respectively.
Expanding this determinant gives the components of A × B.

If A, B and C are coplanar, they all lie in the same plane.
Vector A × B is perpendicular to the plane containing A and B.
Since C is also in that plane, A × B is perpendicular to C as well.

Explanation:

The direction of the cross product vector is given by the right-hand rule: if fingers point along A and curl towards B, then the thumb points in the direction of A × B.

In the right-hand rule, you first point the fingers of your right hand along A.
Then curl the fingers towards B through the smaller angle between them.
Your extended thumb then points in the direction of A × B.

Explanation:

Its magnitude = √(6² + 4²)
= √(36 + 16) = √52

Take A, B and C as i, j and k in a right-handed coordinate system.
We know that i × j = k and k is a unit vector.
Dot product of k with itself is 1.
So (A × B) · C equals 1.

Explanation:

The cross product A × B produces a vector that is perpendicular to both A and B following the right-hand rule. Its magnitude equals |A||B|sinθ, where θ is the angle between A and B.

Explanation:

Using right-hand rule / cyclic order î → ĵ → k̂, we have î × k̂ = –(k̂ × î) = –ĵ.

Cross product of two vectors should give a vector quantity.
Dot product gives a scalar result.
If the answer is scalar while calculating cross product, it means dot product formula was used by mistake.

Cross product satisfies A × (B + C) = A × B + A × C, so distributive law holds.
It also obeys (kA) × B = k(A × B), so scalar multiplication behaves well.
But it is not commutative, because B × A = −(A × B).
So the commutative law does not hold for vector product.

The cross product obeys distributive law over addition.
So A × (B + C) expands as A × B plus A × C.
Therefore A × (B + C) = A × B + A × C.

Swapping the order in a cross product changes the sign of the result.
So B × A equals −(A × B).
This property is called anti-commutativity of the vector product.

Torque measures turning effect of a force about a point.
By definition, torque is given by cross product of position vector r and force F.
So τ = r × F.

Compute A × B = (3i + 4j) × i.
This equals 3(i × i) + 4(j × i).
i × i is zero and j × i = −k.
So A × B = 0 + 4(−k) = −4k.

Cross product magnitude is |A × B| = |A||B| sinθ.
If |A × B| is zero, then either |A| or |B| is zero or sinθ is zero.
Sinθ is zero when vectors are parallel or antiparallel.
So cross product is zero if one vector is zero or the vectors are parallel.

Magnetic force on a moving charge is given by F = q v × B.
Here v is velocity vector and B is magnetic field vector.
This force is a vector product of charge velocity and magnetic field.

Explanation:

Explanation:

= î·[(-2)(-3) – (2·4)]
– ĵ·[1·(-3) – (2·0)]
+ k̂·[1·4 – (-2·0)]

= î·(6 – 8)
– ĵ·(-3 – 0)
+ k̂·(4 – 0)

= (–2)î + 3ĵ + 4k̂

|v| = √[(-2)² + 3² + 4²]
= √[4 + 9 + 16]
= √29

Explanation:

Magnitude of cross product = |A||B|sinθ
Since vectors are perpendicular, θ = 90°, sin90° = 1.
So magnitude
= 4 × 6 × 1 = 24.

Explanation:

Two vectors will be parallel to each other.

Explanation:

Magnitude of cross product = |A||B|sinθ
= 5 × 3 × sin90°
= 15 × 1 = 15.