Arriving at the given solution of a product of three vectors

Question

I've been given three vectors for which a solution is provided, but I've tried the scalar and dot products and am unable to understand figure out what method was used to obtain the answer.

They did mention this:

t . (u x v)

Then they gave that the solution is 34.

But either I don't understand the problem (it's not in English and I'm not very good at the language it's written in) or I just haven't gotten the hang of dot and scalar products properly (I've tried!)

The given vectors whose product is 34, are:

t= 4i - 2j -2k

u= 2i + 4j + 3k

v= i - 5j + 3k

How is t.(u x v) solved to get 34? Or do we not evaluate t.(u x v)?

Answer 1

It is known as triple product, we have the cross product

$$u\times v=\begin{vmatrix}i&j&k\\2&4&3\\1&-5&3\end{vmatrix}=(27,-3,-14)$$

and then by dot product

$$t\cdot ( u\times v)=4\cdot 27-2(-3)-2(-14)=142$$

Answer 2

Your problem is that these are not the dot and scalar product.

First, the dot-product and the scalar product are two names for the same operation. It corresponds to the first operation, . in your expression t.(u x v).

The second operation is called the cross-product. It takes two vectors as input and outputs one vector.

Note that in the end, the expression $t.(u\times v)$ is nothing but the determinant of the three vectors.