Question

What is the cross product of $(2i - 3j + 4k)$ and $(i + j - 7k)$ ?

Answer 1

Hint : Here we will use cross-product. It can be defined as the vector which is the binary operation on two vectors in three dimensions and then will simplify for the resultant required value.

Complete Step By Step Answer:
Let us assume the given vectors as
$\overline a = (2i - 3j + 4k)$ and
$\overline b = (i + j - 7k)$
Now, $$ place the given above two vectors in the determinant form.
\overline a \times \overline b = \left| {\begin{array}{*{20}{c}} i&j;&k; \\\ 2&{ - 3}&4 \\\ 1&1&{ - 7} \end{array}} \right|
Expand the above expression by using the concepts that when we open the minor determinants so formed for respective terms, product of negative term with negative gives positive terms, product of negative term with positive gives negative term and product of positive term with the positive term gives positive.
$\overline a \times \overline b = i(21 - 4) - j( - 14 - 4) + k(2 + 3)$
Terms with the same values and opposite signs cancel each other.
$\overline a \times \overline b = 17i + 18j + 5k$
This is the required solution.

Note :
Remember and apply perfectly the laws of dot product and cross product. Be careful about the sign while doing simplification and remember the golden rules-
Addition of two positive terms gives the positive term
Addition of one negative and positive term, you have to do subtraction and give sign of bigger numbers, whether positive or negative.
Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.
Product of two positive terms gives resultant value in positive.
Product of two negative terms gives resultant value in positive.
Product of one positive and one negative term gives value in negative.