Find the values for $u$ and $v$

Question

Given $u = (2, 4, −3)$,and $v = (3, −1, 7)$.

Find $u\cdot v$ and $u\times v$.

What's the difference between $u\cdot v$ and $u\times v$?

$u\cdot v$ is the dot rule and what is $u\times v$?

Answer 1

"What is the difference between $u \cdot v$ and $u \times v$?"

$u \cdot v$ Is what is called the dot product between $u$ and $v$, and $u \times v$ is called the cross product between $u$ and $v$.

The main difference between the dot product and the cross product is that, is that the if you take the dot product between two vectors you'll always get a number. On the other side, if you take the cross product between two vector you'll get a vector.

The other difference is their meaning:

When you take the dot product between two vectors you get a numbers that basically tells you how pointing in the same direction the vectors are. The formula is the following:

$$u\cdot v=|u||v|\cos(\widehat{uv})$$

Where $\widehat{uv}$ is the angle that the vector $u$ makes with the vector $v$ if you put their tail in the same point.

As you can see with this formula, if the two vectors are perpendicular, this is $\widehat{uv}=90º$, then you'll get that $u\cdot v = 0$, and as I said "When you take the dot product between two vectors you get a numbers that basically tells you how pointing in the same direction the vectors are". If $u\cdot v = 0$ this means that the vectors are not at all pointing in the same direction a.k.a they are perpendicular.

If the angle of the vectors is between $90$ and $0$ you would get that $u\cdot v > 0$ meaning that they are somehow pointing in the same direction. You'll get a bigger value for $u\cdot v$ the smaller the angle between them is, and you have a maximum when the angle is $0$, meaning that they are in the same line!

The cross product is completely different: First of all, you'll always get a vector, not a number. This vector is allways pointing perpendicular to both $u$ and $v$:

The length of the vector $u\times v$ is given by the following formula:

$$|u\times v|=|u||v|\sin(\widehat{uv})$$

As you can see by this formula this is the complete opposite. The length of the vector is $0$ when the angle is $0$ (this meaning that they are in the same line and pointing in the same direction), and as the angle increases, the length increases as well. In fact, the length is maximum when the angle is $90º$ and they are perpendicular and not at all pointing in the same direction. It is the opposite of the dot product!

Another interpretation for the length of the cross product is the following:

The length is the area of the parallelogram that has the vectors $u$ and $v$ as sides! This is another main use of the cross product.

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Now on how to calculate those: The formulas are pretty simple:

Let $A = (a_1,a_2,a_3)$ and $B=(b_1,b_2,b_3)$ be vectors. Then:

$$A\cdot B = a_1 \cdot b_1 + a_2 \cdot b_2 +a_3 \cdot b_3 $$

And for the cross product:

$$A \times B = \det \left|\begin{matrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{matrix} \right|$$

Where $\hat i, \hat j , \hat k $ are the basis vectors in $\Bbb R^3$