Find the cross products 𝐮 ×𝐯 and v ×u for the following vectors 𝐮 and 𝐯. 𝐮=3 𝐢-𝐣-2 𝐤, 𝐯=𝐢+3 𝐣-2

Find the cross products 𝐮 ×𝐯 and v ×u for the following...

Find the cross products $\mathbf{u} \times \mathbf{v}$ and v $\times$ u for the following vectors $\mathbf{u}$ and $\mathbf{v}$.
$$\mathbf{u}=3 \mathbf{i}-\mathbf{j}-2 \mathbf{k}, \mathbf{v}=\mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$$

Key Concepts

Determinant Method for Cross Product

A common method to compute the cross product is by using the determinant of a 3×3 matrix whose first row contains the standard unit vectors and whose second and third rows contain the components of the two vectors. This method provides a systematic way to evaluate the cross product by computing the determinants of the 2×2 submatrices.

Anticommutativity

An important property of the cross product is that it is anticommutative. This means that if you swap the order of the two vectors, the result is the negative of the original cross product; that is, u × v = - (v × u). This property is critical in ensuring consistency with the right-hand rule and simplifying calculations in vector algebra.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in a vector which is perpendicular to both of the original vectors. Its magnitude represents the area of the parallelogram formed by the two vectors, and it plays an important role in physics and engineering, particularly in the analysis of rotational forces and torque.

Transcript

00:01 Hello everybody.

00:02 In this video, i'm going to be showing you how to solve exercise 27 in chapter 13, section 4 of calculus early transcendentals.

00:11 Now this problem gives us two vectors, u, which is 3i minus j minus 2k, and v, which is i plus 3j minus 2k, where i, j, j, and k are the standard unit vectors.

00:26 And with these two vectors, they want us to find the cross products, u cross v, as well as v, cross u now there are multiple ways to calculate u cross v but i'm going to be using the determinant method as described in theorem 13 .6 of this chapter to do this we want to create a three by three matrix whose top row contains the three standard unit vectors whose second row contains the units of u three negative 1 and negative 2, the coefficients of i, j, and k.

01:10 And the bottom row has the components of v, 1, 3, and negative 2.

01:18 To calculate this determinant, what we do is we first take i, and we multiply it by the determinant of its corresponding sub -matrix, which we find by deleting its row and column.

01:30 And so we're left with this bottom 2x2 matrix on the right here.

01:35 So we have the determinant of negative 1, negative 2, 3, negative 2 times i.

01:45 From this, we subtract a similar quantity for j.

01:48 We delete its row and column, and end up with these two subcolums on the left and right, getting 3, negative 2, 1, negative 2 times j.

02:02 And then to this, we add the similar case for k.

02:05 Deleting these, we end up with the sub matrix on the bottom left...

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