Find (a) 𝐮 ×𝐯, (b) 𝐯 ×𝐮, and (𝐜) 𝐯 ×𝐯. 𝐮=3 𝐢+5 𝐤 𝐯=2 𝐢+3 𝐣-2 𝐤

Name: Problem 8, Chapter 11: Calculus Early Transcendental Functions
Uploaded: 2021-08-03T14:54:08-08:00
Duration: 4 min 12 s
Channel: Audrey Fong
Description: Find (a) 𝐮 ×𝐯, (b) 𝐯 ×𝐮, and (𝐜) 𝐯 ×𝐯. 𝐮=3 𝐢+5 𝐤 𝐯=2 𝐢+3 𝐣-2 𝐤

Find (a) 𝐮 ×𝐯, (b) 𝐯 ×𝐮, and (𝐜) 𝐯 ×𝐯. 𝐮=3 𝐢+5 𝐤 ...

Find (a) $\mathbf{u} \times \mathbf{v}$, (b) $\mathbf{v} \times \mathbf{u},$ and $(\mathbf{c}) \mathbf{v} \times \mathbf{v}$.
$$\begin{array}{l}\mathbf{u}=3 \mathbf{i}+5 \mathbf{k} \\\mathbf{v}=2 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}\end{array}$$

Key Concepts

Vector Cross Product

The cross product is a binary operation on two vectors in three-dimensional space that results in another vector which is perpendicular to both original vectors. It is defined using the determinants of a matrix constructed from the unit vectors and the components of the original vectors, and its magnitude represents the area of the parallelogram spanned by the two vectors.

Anti-Commutativity

The cross product is anti-commutative, meaning that swapping the order of the vectors negates the resulting vector. In mathematical terms, for any two vectors u and v, u × v = - (v × u). This property is important when rearranging terms in vector calculations.

Zero Product of Identical Vectors

The cross product of any vector with itself is always the zero vector. This is because the angle between the vector and itself is zero, and since the sine of zero is zero, the magnitude of the cross product becomes zero. This property confirms that there is no vector perpendicular to a vector itself.

Transcript

00:01 In this question, we are given vector u to be this and vector v to be this.

00:05 We want to explore u cross v, v cross u and b cross v.

00:12 Now before we do that, let's look at some properties of cross vectors.

00:17 Now, when a cross b is defined to be magnitude of a times magnitude of b, sine theta, and multiply to a normal vector, that is the direction vector for this a cross b.

00:32 And this a cross b is perpendicular to both a and perpendicular to b at the same time.

00:38 Now, one property of a cross b is not commutative, so a cross b is equals to minus b cross a.

00:50 So that's one of property.

00:52 Another property, if a is parallel to b, the vectors, then a cross b will be the zero vector.

01:04 Now this is easy because if a is parallel to b, the angle between them titter will be 0.

01:11 So sign 0 will be 0.

01:13 So the whole expression here on the right side will just become 0 vector.

01:18 Now, i mentioned vector because there is still a vector over here, a direction here.

01:23 Okay.

01:24 So now let's explore u cross b.

01:26 Now, it's easier to express u and b in column vector.

01:30 So u will be 305.

01:34 And v will be 2, 3 minus 2.

01:39 So v cross u will be 305 cross 2 3 -3 -1.

01:48 Now i'll be using the cover -up method.

01:51 Now, when i want to get the answer to my first row, i will cover up my first row.

01:57 So that means i don't look at my first row.

02:00 I only look at the second and third row...

Please give Ace some feedback