Prove the following identities. (a) (𝐮+k 𝐯) ×𝐯=𝐮 ×𝐯 (b) 𝐮 ·(𝐯 ×𝐳)=-(𝐮 ×𝐳) ·𝐯

Name: Problem 34, Chapter 3: Elementary Linear Algebra: Applications Version
Uploaded: 2021-05-08T06:26:43-08:00
Duration: 3 min 20 s
Channel: Manisha Sarker
Description: Prove the following identities. (a) (𝐮+k 𝐯) ×𝐯=𝐮 ×𝐯 (b) 𝐮 ·(𝐯 ×𝐳)=-(𝐮 ×𝐳) ·𝐯

Prove the following identities. (a) (𝐮+k 𝐯) ×𝐯=𝐮 ×𝐯 (b) 𝐮...

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Key Concepts

Distributivity of the Cross Product

The cross product is linear in each of its arguments, meaning that it is distributive over vector addition. This property allows one to expand expressions like (u + kv) x v into u x v + k(v x v), which is essential for simplifying and proving vector identities.

Zero Result of Self Cross Product

A key property of the cross product is that any vector crossed with itself yields the zero vector. This is because the angle between the vector and itself is zero, and the sine of zero is zero. This concept is crucial for simplifying terms such as v x v in vector proofs.

Anti-Commutativity of the Cross Product

The cross product is anti-commutative, meaning that u x v is equal to -(v x u). This property is fundamental when manipulating and rearranging cross product expressions and is particularly useful in proofs involving sign changes.

Properties of the Scalar Triple Product

The scalar triple product, defined as u · (v x w), possesses a cyclic permutation property and a sign change upon swapping two of its vectors. This concept helps in proving identities such as u · (v x z) = -(u x z) · v by demonstrating how reordering the vectors affects the overall sign of the product.

Transcript

00:01 In this problem, we're going to use the concepts and properties of cross products in order to prove certain identities.

00:08 Now, in the first problem, the left -hand side of the identity is u plus kv cross -v, where u and v are vectors and k is a scalar.

00:26 Now, since the cross -product follows the distributive property, this will be equals to u -cross -v.

00:34 Plus kv cross v.

00:38 So this can be written as u cross v by leaving the first term unchanged.

00:42 And as for the second term, since k is a scalar, we can write this as k times v cross v.

00:52 The first term will be left unchanged as u cross v plus k times v cross v, which will be the zero vector since the cross product of any vector with itself is always going to be the zero vector.

01:06 So this will be u cross v plus the zero vector because any scalar times the zero vector will be the zero vector.

01:15 And since adding the zero vector to any vector will result in the original vector, this will just be equals to u cross v and this is the right hand side of the identity and hence the identity has been proved.

01:32 As for the second identity, the left -hand side is u.

01:43 .v cross z.

01:46 Now this is a scalar triple product.

01:49 So we can write this as a determinant of order 3, where the first row has the components of the vector u, say u1, u2, u3...

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