Question

Calculu&sVectors: If a = 3i - 5j + 7k , b = −5i + 5j + 2k and c = 2i − j + 3k find:
a. 2a + b
b. a unit vector in the direction of c
c. a • b
d. c × a

Answer 1

\[\vec{a}=3\vec{i}-5\vec{j}+7\vec{k}\\
\vec{b}=-5\vec{i}+5\vec{j}+2\vec{k}\\
\vec{c}=2\vec{i}-\vec{j}+3\vec{k}\]
(a) \(2\vec{a}+\vec{b}\): Multiply the components of \(\vec{a}\) by 2, then add the components of \(2\vec{a}\) and \(\vec{b}\).
\[2\vec{a}=2(3\vec{i}-5\vec{j}+7\vec{k})=6\vec{i}-10\vec{j}+14\vec{k}\]
I'll leave the addition to you.

Answer 2

(b) unit vector in direction of \(\vec{c}\): This is given by \(\dfrac{\vec{c}}{|\vec{c}|}\), where \(|\vec{c}|\) is the magnitude of \(\vec{c}\).

Answer 3

(c) \(\vec{a}\cdot\vec{b}\): Given two vectors \(\vec{a}=a_1\vec{i}+a_2\vec{j}+a_3\vec{k}\) and \(\vec{b}=b_1\vec{i}+b_2\vec{j}+b_3\vec{k}\), the dot product of \(\vec{a}\) and \(\vec{b}\) is
\[\vec{a}\cdot\vec{b}=\sum_{i=1}^3a_ib_i=a_1b_1+a_2b_2+a_3b_3\]

Answer 4

(d) \(\vec{c}\times\vec{a}\): Find the following determinant:
\[\begin{vmatrix}
\vec{i}&\vec{j}&\vec{k}\\
2&-1&3\\
3&-5&7
\end{vmatrix}\]