2.3
Your guide: Sir HamzaBelieves every formula has a story, and units never lie.
Pushing a box needs force. Work depends on how much of the force is along the path.
That idea is the dot product. Its partner is the cross product.
There are two vector products: the scalar product and the vector product.
If the answer is a scalar, it is the scalar product. If it is a vector, it is the vector product.
A · B = AB cos θ (Eq. 2.6). Here θ is the angle between them.
It is A times the component of B along A.
Work is a scalar product: F · d = Fd cos θ.
Five properties from the book:
1. A · B = B · A. The order does not matter.
2. For perpendicular vectors, A · B = AB cos 90° = 0.
3. Parallel (0°): A · B = AB. Antiparallel (180°): A · B = −AB.
4. A · A = A2.
5. A · B = AₓBₓ + A_yB_y + A_zB_z (Eq. 2.7). So cos θ = (AₓBₓ + A_yB_y + A_zB_z) ÷ AB (Eq. 2.8).
A × B = AB sin θ n̂ (Eq. 2.9). n̂ is a unit vector perpendicular to the plane of A and B.
Right hand rule: turn A into B by the smaller angle. Curl the fingers that way. The thumb gives the direction.
A × B = − B × A (Eq. 2.10). So the cross product is not commutative.
For perpendicular vectors (90°) the size is largest: AB.
For parallel (0°) or antiparallel (180°) vectors it is a null vector. So A × A = 0.
The size |A × B| equals the area of the parallelogram on A and B.
Examples: torque τ = r × F and magnetic force F = q(v × B).
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Pushing a box needs force. Work depends on how much of the force is along the path.
That idea is the dot product. Its partner is the cross product.
There are two vector products: the scalar product and the vector product.
If the answer is a scalar, it is the scalar product. If it is a vector, it is the vector product.
A · B = AB cos θ (Eq. 2.6). Here θ is the angle between them.
It is A times the component of B along A.
Work is a scalar product: F · d = Fd cos θ.
Five properties from the book:
1. A · B = B · A. The order does not matter.
2. For perpendicular vectors, A · B = AB cos 90° = 0.
3. Parallel (0°): A · B = AB. Antiparallel (180°): A · B = −AB.
4. A · A = A2.
5. A · B = AₓBₓ + A_yB_y + A_zB_z (Eq. 2.7). So cos θ = (AₓBₓ + A_yB_y + A_zB_z) ÷ AB (Eq. 2.8).
A × B = AB sin θ n̂ (Eq. 2.9). n̂ is a unit vector perpendicular to the plane of A and B.
Right hand rule: turn A into B by the smaller angle. Curl the fingers that way. The thumb gives the direction.
A × B = − B × A (Eq. 2.10). So the cross product is not commutative.
For perpendicular vectors (90°) the size is largest: AB.
For parallel (0°) or antiparallel (180°) vectors it is a null vector. So A × A = 0.
The size |A × B| equals the area of the parallelogram on A and B.
Examples: torque τ = r × F and magnetic force F = q(v × B).
Q1. What are A · B and |A × B| for perpendicular vectors of sizes A and B?
A · B = AB cos 90° = 0. |A × B| = AB sin 90° = AB.
✗ “A × B = B × A.”
✓ The cross product changes sign: A × B = − B × A.
✗ “The dot product of two vectors is a vector.”
✓ It is a scalar. That is why work is a scalar.
1. A · B for perpendicular vectors is:
(b) cos 90° = 0.
2. A × A is:
(b) sin 0° = 0.
3. Which is a scalar product?
(a) The book writes work as F · d = Fd cos θ.
4. For antiparallel vectors, A · B equals:
(c) cos 180° = −1.
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