Which of the following are always true, and which are not...

Which of the following are always true, and which are not always true? Give reasons for your answers. \begin{array} { l } { \text { a. } \mathbf { u } \cdot \mathbf { v } = \mathbf { v } \cdot \mathbf { u } } \\ { \text { b. } \mathbf { u } \times \mathbf { v } = - ( \mathbf { v } \times \mathbf { u } ) } \\ { \text { c. } \quad ( - \mathbf { u } ) \times \mathbf { v } = - ( \mathbf { u } \times \mathbf { v } ) } \\ { \text { d. } ( c \mathbf { u } ) \cdot \mathbf { v } = \mathbf { u } \cdot ( c \mathbf { v } ) = c ( \mathbf { u } \cdot \mathbf { v } ) \quad \text { (any number } c ) } \\ { \text { e. } c ( \mathbf { u } \times \mathbf { v } ) = ( c \mathbf { u } ) \times \mathbf { v } = \mathbf { u } \times ( c \mathbf { v } ) \quad \text { (any number } c ) } \\ { \text { g. } \quad \mathbf { u } \cdot \mathbf { u } = \mathbf { u } = 0 } \\ { \text { h. } ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = \mathbf { v } \cdot ( \mathbf { u } \times \mathbf { v } ) } \end{array}

Which of the following are always true, and which are not always true? Give reasons for your answers.
\begin{array} { l } { \text { a. } \mathbf { u } \cdot \mathbf { v } = \mathbf { v } \cdot \mathbf { u } } \\ { \text { b. } \mathbf { u } \times \mathbf { v } = - ( \mathbf { v } \times \mathbf { u } ) } \\ { \text { c. } \quad ( - \mathbf { u } ) \times \mathbf { v } = - ( \mathbf { u } \times \mathbf { v } ) } \\ { \text { d. } ( c \mathbf { u } ) \cdot \mathbf { v } = \mathbf { u } \cdot ( c \mathbf { v } ) = c ( \mathbf { u } \cdot \mathbf { v } ) \quad \text { (any number } c ) } \\ { \text { e. } c ( \mathbf { u } \times \mathbf { v } ) = ( c \mathbf { u } ) \times \mathbf { v } = \mathbf { u } \times ( c \mathbf { v } ) \quad \text { (any number } c ) } \\ { \text { g. } \quad \mathbf { u } \cdot \mathbf { u } = \mathbf { u } = 0 } \\ { \text { h. } ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = \mathbf { v } \cdot ( \mathbf { u } \times \mathbf { v } ) } \end{array}

Key Concepts

Dot Product

The dot product is an operation on two vectors that returns a scalar. It is defined as the sum of the products of the corresponding components of the two vectors and has key properties such as commutativity and distributivity over addition. Additionally, the dot product of a vector with itself gives the square of its norm, which is zero only when the vector is the zero vector.

Commutative Property of the Dot Product

A fundamental property of the dot product is its commutativity, meaning that the order of the vectors does not affect the outcome; u • v is always equal to v • u. This property is essential in many proofs and simplifications in vector algebra.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that returns a vector perpendicular to both of the original vectors. Its magnitude is equal to the area of the parallelogram defined by the two vectors, and it is not commutative but rather anti-commutative.

Anti-Commutative Property of the Cross Product

For the cross product, switching the order of the vectors reverses the sign of the resulting vector; that is, u × v equals -(v × u). This property is crucial in understanding the orientation of the resultant vector in relation to the original vectors.

Scalar Multiplication Properties

Both the dot product and the cross product have linearity properties with respect to scalar multiplication. This means that scaling one operand by a constant scales the result by the same constant, and the constant can be factored out of the operation. These properties ensure that operations are consistent with the structure of vector spaces.

Orthogonality of the Cross Product

An important aspect of the cross product is that its resulting vector is orthogonal to both of the original vectors. This property leads to useful relationships, such as the fact that the dot product of the cross product with either original vector is zero, thereby simplifying many vector calculations and proofs.

Transcript

00:01 So here we're given a bunch of properties of vectors and related to dot and cross products, and we're asked which of these are always true and which of them might be false sometimes.

00:14 So let's go through the list here.

00:17 Are, is the dot product commutative? yes, that's always true.

00:23 Just because, you know, a dot product is just multiplication and addition of numbers, and those operations are commutative.

00:34 Okay, so what about property b? the cross product is not commutative.

00:40 If you switch them, it's actually negative.

00:43 And this is also true.

00:45 And this sort of makes sense if you think of it in terms of the right -hand rule.

00:49 So let's say if i have, these are my vectors u and v, and by the right -hand rule, u -cross -v is going to be something that's perpendicular to both of them, and pointing in the direction where if i had not a number, my fingers pointed towards you with my palm facing v, my thumb is pointing in the direction of u cross v.

01:13 So if i do that in reverse, if i first point my fingers in the direction of v with my palm facing u, then my thumb is going to be pointing in the opposite direction.

01:24 They have the same magnitude.

01:26 They're just in the opposite direction.

01:28 So that's why property b is true.

01:32 And similarly, you can draw another picture and see why property c should be true.

01:37 So you can either draw a picture or just check the formula.

01:42 Okay, and then properties d and e are similar, and they both are also true.

01:50 And this just follows from the distributive law, which is something that's true for both the scalar product and the cross product...

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