3]Dot & Cross Product with Examples - Vector Analysis - GATE Engineering Mathematics

Summary

In this video, the presenter explains two fundamental vector operations: the dot product and the cross product. The dot product, yielding a scalar value, is defined by the product of the magnitudes of two vectors and the cosine of the angle between them. In contrast, the cross product results in a vector, derived from the magnitudes and the sine of the angle, and is oriented perpendicular to the plane of the two vectors. The video further illustrates these concepts with examples, highlights their properties, and introduces the scalar and vector triple products, emphasizing their applications in physics and engineering.

Takeaways

• 😀 The dot product of two vectors is calculated using the formula: A · B = |A| |B| cos(θ), resulting in a scalar quantity.
• 😀 The dot product is commutative, meaning A · B = B · A, and it is also distributive over vector addition.
• 😀 If the angle between two vectors is 0 degrees, their dot product equals the product of their magnitudes.
• 😀 The cross product of two vectors is calculated as A × B = |A| |B| sin(θ) n, resulting in a vector quantity.
• 😀 The cross product is not commutative; A × B = - (B × A). This means the order of vectors matters.
• 😀 The right-hand rule is used to determine the direction of the resultant vector in a cross product.
• 😀 The scalar triple product combines the dot product and cross product, resulting in a scalar value.
• 😀 The vector triple product involves the cross product of two vectors and results in a vector output.
• 😀 The dot product provides a measure of how much one vector extends in the direction of another, while the cross product indicates the area spanned by the two vectors.
• 😀 Determinants can be used to simplify calculations for the cross product, allowing for easy representation of 3D vector relationships.

Q & A

What is the dot product of two vectors?

The dot product is calculated by multiplying the magnitudes of two vectors and the cosine of the angle between them, resulting in a scalar quantity.

Why does the dot product yield a scalar quantity?

The dot product gives a scalar quantity because it only considers the magnitude of the projection of one vector onto another, without any direction.

What happens to the dot product when the angle between two vectors is 90 degrees?

When the angle between two vectors is 90 degrees, the cosine of 90 is 0, resulting in a dot product of 0.

What is the cross product of two vectors?

The cross product is calculated by multiplying the magnitudes of two vectors and the sine of the angle between them, resulting in a vector quantity.

How is the direction of the cross product determined?

The direction of the cross product is determined using the right-hand rule, where the resulting vector is perpendicular to the plane formed by the two original vectors.

What does it mean if the cross product results in a negative vector?

A negative vector in the cross product indicates that the direction of the resultant vector is opposite to that obtained using the right-hand rule, typically resulting from clockwise rotation.

Can the dot product and cross product be commutative?

The dot product is commutative (A·B = B·A), while the cross product is not commutative (A × B = -B × A).

What is the scalar triple product?

The scalar triple product is defined as the dot product of one vector with the cross product of the other two vectors, yielding a scalar output.

What is the vector triple product?

The vector triple product is the cross product of one vector with the cross product of the other two vectors, yielding a vector output.

How are determinants used in calculating cross products?

Determinants are used to organize the coefficients of the vectors in a 3x3 matrix format, which facilitates the calculation of the cross product using the determinant method.