Let 𝐯 and 𝐰 be nonzero vectors. Show that the vectors 𝐰 𝐯+𝐯 𝐰 and 𝐰 𝐯-𝐯 𝐰 are orthogonal.

Name: Problem 58, Chapter 10: Calculus, Early Transcendentals
Uploaded: 2021-08-02T20:52:40-08:00
Duration: 3 min 50 s
Channel: Monica Miller
Description: Let 𝐯 and 𝐰 be nonzero vectors. Show that the vectors 𝐰 𝐯+𝐯 𝐰 and 𝐰 𝐯-𝐯 𝐰 are orthogonal.

step 1 Okay, let V and W be non -zero vectors. And we are going to show that the magnitude of W times v

Let 𝐯 and 𝐰 be nonzero vectors. Show that the vectors 𝐰 𝐯+𝐯...

Key Concepts

Vector Norm

A vector norm is a function that assigns a non-negative length or size to each vector in a vector space, typically measuring the distance from the origin to the point represented by the vector. Norms obey properties such as positivity, scalability, and the triangle inequality, and are fundamental in expressing geometric concepts and scaling vectors.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is defined as the sum of the products of the corresponding entries, providing a measure of how much two vectors are aligned. This scalar is also crucial in computing angles between vectors as well as establishing conditions like orthogonality.

Orthogonality

Orthogonality of vectors occurs when their dot product is zero, indicating that the vectors are perpendicular to each other in the context of Euclidean space. This concept is pivotal in geometry, linear algebra, and many applied fields where perpendicularity leads to simplification in calculations and analysis.

Distributive and Commutative Properties of the Dot Product

The dot product adheres to properties such as distributivity over vector addition and commutativity. The distributive property (u • (v + w) = u • v + u • w) allows for the expansion of expressions involving sums of vectors, while commutativity (u • v = v • u) simplifies the manipulation of algebraic expressions, both of which are essential in proving relationships like orthogonality in vector combinations.

Transcript

00:01 Okay, let v and w be non -zero vectors.

00:04 And we are going to show that the magnitude of w times v plus the magnitude of v times the vector w is orthogonal to the magnitude of w times v minus the magnitude of v times w.

00:21 Okay, so this looks a little complex to start.

00:24 So let's just kind of break down some pieces.

00:29 We're definitely going to have to define some vectors here.

00:33 So let's make vector v, a, comma, b, and vector w, c, comma, d.

00:39 So we're really just going to start this out by stating, you know, vector w times, or the magnitude of w, and then putting our vector v in, plus the magnitude of v and putting our vector w in.

00:56 So sometimes you don't necessarily know where you, you're going as you start, you just kind of start finding things to put in.

01:04 So at this point, i am now putting in my magnitudes.

01:09 So i'm squaring and summing all my components with a square root over them.

01:15 Okay, so that has given me, you know, pretty much as far as i want to go with that first statement.

01:22 So i'm going to do the same thing with the second statement.

01:25 Again, i have now what i consider to be my vectors.

01:36 Well, let's go ahead and multiply the i component in.

01:40 So i have my a times that magnitude of w plus the c times the magnitude of v that i can then put together.

01:50 So i'm just cleaning up kind of the statement above, but i'm separating my i components from my j components...

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