Given three vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) show that in intrinsic representation the magnitude: \(V=(\mathbf{a} \times\) b) - c corresponds to the volume of the parallelepiped defined by the three vectors. Calculate the volume in the case of the vectors \(\mathbf{a}, \mathbf{b}\) defined in the Question 8 , and \(\mathbf{c}=(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}})\).

Question 8

Find the vector \(\mathbf{c}\) which added to the vectors \(\mathbf{a}=(4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})\) and \(\mathbf{b}=(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})\) gives as a resultant \(\mathbf{r}=(4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+1 \hat{\mathbf{k}})\)