Vectors - Geometric interpretation of cross product
In this lecture, we will explore the geometric interpretation of the cross product.
The cross product of two vectors gives us a vector that is orthogonal to the two original vectors.
It is denoted by the symbol × or ⨯.
Geometric Interpretation
The magnitude of the cross product of two vectors A and B is given by:
|A × B| = |A| |B| sinθ
θ is the angle between the two vectors.
The direction of the cross product is determined by the right-hand rule:
If you extend your right hand’s fingers in the direction of vector A and then curl them towards vector B, your thumb points in the direction of the cross product.
Properties of Cross Product
The cross product is not commutative:
A × B = - (B × A)
The length of the cross product is:
|A × B| = |A| |B| sinθ
The direction of the cross product is orthogonal to both A and B.
If A and B are parallel, the cross product is zero:
A × B = 0
If A and B are perpendicular, the magnitude of the cross product is:
|A × B| = |A| |B|
Example 1
Let A = i + 2j and B = 3i + k be two vectors.
Find the cross product of A and B.
Solution:
(i × 3i) + (i × k) + (2j × 3i) + (2j × k)
= 0 + (-k) + (6k) + 0
= 5k
Therefore, A × B = 5k.
Example 2
Given two vectors A = 2i - 3j + 4k and B = 5i + 2j - k, find the angle between them.
Solution:
|A × B| = |A| |B| sinθ
|A × B| = sqrt(29) sqrt(30) sinθ
|A × B| = 5 sqrt(2) sqrt(3) sinθ
Since A × B = 0 (cross product is zero for parallel vectors), sinθ = 0.
Therefore, θ = 0 degrees.
Example 3
Let A = i + j + k and B = 2i - j - 2k be two vectors.
Find the vector that is orthogonal to both A and B.
Solution:
A × B = (i × (2i - j - 2k)) + (j × (2i - j - 2k)) + (k × (2i - j - 2k))
= (-5i - 4j + 3k)
Therefore, the vector orthogonal to both A and B is -5i - 4j + 3k.
Note: Slide numbering is not allowed, so the slides will be labeled with headings only. Also, expressions and equations may not be accurately represented in this text format.
Slide 11: Properties of Cross Product (continued)
The cross product is distributive:
A × (B + C) = (A × B) + (A × C)
The cross product satisfies the scalar triple product rule:
A · (B × C) = B · (C × A) = C · (A × B)
This rule is useful in solving problems involving vectors.
Slide 12: Geometric Interpretation (continued)
The cross product of two vectors can also be interpreted as the area of the parallelogram formed by those vectors.
The magnitude of the cross product is equal to the area of the parallelogram.
Slide 13: Example 4
Given two vectors A = 2i - 3j + k and B = 4i - 2j + 3k, find the area of the parallelogram formed by these vectors.
Therefore, the area of the parallelogram formed by A and B is sqrt(38) square units.
Slide 14: Example 5
Find the position vector of the point where the line defined by A = 2i - 3j + 4k + λ(i - 2j + k) intersects the plane defined by B = 3i - j + 2k + μ(2i + j - k) + v(i + j - 3k).
Solution:
The normal vector to the plane is the cross product of the two direction vectors of the plane, B and C.
N = B × C = (9i + 5j + 9k).
The point where the line intersects the plane can be found using the formula:
P = A + λ(N dot (D - A))/ (N dot (D - A)),
where D is a point on the line.
Now, substitute the values of A, B, C, and D to find the position vector of the point of intersection.
Slide 16: Application - Torque
In physics, the cross product of a force vector and a displacement vector gives us the torque or moment of the force.
Torque is a measure of the rotational effect of a force on an object.
Torque = r × F
r is the displacement vector from the point of rotation to the point of application of the force.
F is the force vector.
Slide 17: Application - Torque (continued)
The magnitude of the torque is given by:
|Torque| = |r| |F| sinθ
θ is the angle between the displacement and force vectors.
The direction of the torque is given by the right-hand rule:
If you extend your right hand’s fingers in the direction of the displacement vector and then curl them towards the force vector, your thumb points in the direction of the torque.
Slide 18: Summary of Cross Product
The cross product of two vectors has a geometric interpretation as a vector orthogonal to both vectors.
It can also be interpreted as the area of the parallelogram formed by the two vectors.
The cross product has various properties, such as distributivity and scalar triple product rule.
It is useful in solving problems involving torque and finding points of intersection between lines and planes.
Slide 19: Summary of Geometric Interpretation
The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors.
The direction of the cross product is orthogonal to both vectors.
The right-hand rule can be used to determine the direction of the cross product.
Slide 20: Conclusion
The geometric interpretation of the cross product is essential to understand its properties and applications.
The cross product provides a useful tool in various fields like physics and engineering.
Practice solving problems involving cross product to enhance your understanding and problem-solving skills.
Vectors - Geometric interpretation of cross product
Slide 21:
The cross product can also be used to find the angle between two vectors.
The dot product of the two vectors can be used in combination with the cross product to find the angle.
Vectors - Geometric interpretation of cross product In this lecture, we will explore the geometric interpretation of the cross product. The cross product of two vectors gives us a vector that is orthogonal to the two original vectors. It is denoted by the symbol × or ⨯.