Problem 2 compute the following in this basis set hatk hatj...

Name: problem 2 compute the following in this basis set hatk hatj hatig a hati cdot 4hati b 4hatj cdot 4hati c 2hatk cdot hati d hatk times 4hati e hati times hati e hatj times hati 23208
Uploaded: 2025-02-12T21:12:02-08:00
Duration: 7 min 20 s
Channel: Eduard Sanchez
Description: problem 2 compute the following in this basis set hatk hatj hatig a hati cdot 4hati b 4hatj cdot 4hati c 2hatk cdot hati d hatk times 4hati e hati times hati e hatj times hati 23208

Problem 2: Compute the following in this basis set $\{\hat{k}, \hat{j}, \hat{i}\}_{G}$: a. $\hat{i} \cdot 4\hat{i}$ b. $4\hat{j} \cdot 4\hat{i}$ c. $2\hat{k} \cdot \hat{i}$ d. $\hat{k} \times 4\hat{i}$ e. $\hat{i} \times \hat{i}$ e. $\hat{j} \times \hat{i}$

Transcript

00:01 Hi there, so for this problem, we need to solve the following operations by the figures that we are, by the vectors that we are provided in here.

00:12 So for example, the first expression that we need to calculate, it will be a dot product with f plus two times c.

00:26 So first of all, let's write these three vectors in a vector form.

00:29 So for example for the vector a, as you can see from the figure, the vector a is in the first quadrant.

00:37 So both the x component and the y component are positive and the angle is with respect to the horizontal.

00:46 So when we calculate the x component, that will be given by the cosine of that angle.

00:51 So that will be the magnitude that we are given for this, that is 10, this times the cosine of 30 degrees.

00:57 This in the x -derption plus 10 times the sign of 30 degrees, that in the wide direction.

01:04 Now, once we have this, also we need to write the vector f.

01:10 As you can see from the figure, the vector f is downward and to the left, so that means that it is in the third quadrant, so both components are negative.

01:21 So the magnitude and the angle is given with respect to the horizontal.

01:24 So then the x component is as well given by the cosine of that angle.

01:31 So that will be minus its magnitude times the cosine of 30 degrees.

01:38 Then this in the x eruption plus 20 times the sign of 30 degrees.

01:44 This is in the wide eruption.

01:45 Now for the better, as you can see from the figure, it's in the fourth quadrant.

01:49 So the white component is negative and the x company is positive.

01:54 Again, the angle is given good respect to the horizontal, so then this will be the edge component positive.

02:01 The magnitude is 12, 12 times the cosine of 60 degrees.

02:05 Then this minus 12 times the sign of 60 degrees, this in the y direction.

02:12 Okay, so once we have this, what we need to do is to do this operation in here.

02:17 I'm going to lift that to you, but you just need to see multiplied, each term by two.

02:27 So then for example, two times this, what are you need to change our this value? so that will give us 24, 24, because that is 2 times 12.

02:36 And then apt them these two together.

02:40 And then that result, you do the dot product with a.

02:44 Remember that when we do the product, for example, product of a time a dot product with some vector, let's say f, that will be the x component of a times the x component of b plus the y component of a times the y component of b.

03:03 So first you need to do f plus 2 times c and then that dot product with a.

03:09 Now let's pass now to part e of this problem, which is the dot product between i and this dot product with b.

03:17 First let's write what it is b.

03:19 B, as you can see from the figure, it is in the first quadrant...

Question

Problem 2: Compute the following in this basis set $\{\hat{k}, \hat{j}, \hat{i}\}_{G}$: a. $\hat{i} \cdot 4\hat{i}$ b. $4\hat{j} \cdot 4\hat{i}$ c. $2\hat{k} \cdot \hat{i}$ d. $\hat{k} \times 4\hat{i}$ e. $\hat{i} \times \hat{i}$ e. $\hat{j} \times \hat{i}$

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