Question

If u = i + 3j - 2k, and v = 4i - 2j - 4k. Find ( 2u + v ) ( u - 2v ).

Answer 1

Question 31562

$\vec{u} (1;3; - 2),\vec{v} (4; - 2; - 4)$

Vector $(a_{1};a_{2};a_{3})$ is multiplied by scalar $\lambda$ by following rule: $\lambda (a_1;a_2;a_3) = (\lambda a_1;\lambda a_2;\lambda a_3)$

Vectors are added(or subtracted) by components: $(a_{1};a_{2};a_{3}) + (b_{1};b_{2};b_{3}) = (a_{1} + b_{1};a_{2} + b_{2};a_{3} + b_{3})$

Using these rules, obtain $2\vec{u} + \vec{v} = (6; 4; -8)$ and $\vec{u} - 2\vec{v} = (-7; 7; 6)$ .

The dot product of two vectors $\vec{a}(a_1; a_2; a_3)$ and $\vec{b}(b_1; b_2; b_3)$ is $\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$ .

Hence, $(2\vec{u} +\vec{v})\cdot (\vec{u} -2\vec{v}) = -7\cdot 6 + 4\cdot 7 + 6\cdot (-8) = -62$