QuestionGiven $\quad u=1+2 j+k, v=3 i+j+2 k$ , find the following cross products:

Studdy Solution

STEP 1

What is this asking? We're asked to find the cross product of two 3D vectors, $u$ and $v$ . Watch out! Remember, the cross product isn't commutative! $u \times v$ is not the same as $v \times u$ .
The order matters!

STEP 2

1. Set up the vectors
2. Calculate the cross product

STEP 3

Let's rewrite our vectors $u$ and $v$ in a way that makes it super easy to calculate the cross product.
We'll use the standard basis vectors $i$ , $j$ , and $k$ .

STEP 4

We have $u = 1i + 2j + 1k$ and $v = 3i + 1j + 2k$ .
Notice how we explicitly wrote the coefficients of $i$ , $j$ , and $k$ for both vectors.
This will be super helpful in the next step!

STEP 5

Now, let's calculate $u \times v$ !
Remember the formula for the cross product: $u \times v = \begin{vmatrix} i & j & k \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$ where $u = u_1 i + u_2 j + u_3 k$ and $v = v_1 i + v_2 j + v_3 k$ .

STEP 6

Plug in our values for $u$ and $v$ : $u \times v = \begin{vmatrix} i & j & k \\ 1 & 2 & 1 \\ 3 & 1 & 2 \end{vmatrix}$

STEP 7

Compute the determinant: $u \times v = i(2 \cdot 2 - 1 \cdot 1) - j(1 \cdot 2 - 1 \cdot 3) + k(1 \cdot 1 - 2 \cdot 3)$ $u \times v = i(4 - 1) - j(2 - 3) + k(1 - 6)$ $u \times v = 3i + j - 5k$

STEP 8

So, the cross product of $u$ and $v$ is $3i + j - 5k$ !

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