Answer:
a) [tex]\:<\:2,1,-3\:>\:=\sqrt{14}\cdot \frac{\:<\:2,1,-3\:>\:}{\sqrt{14} }[/tex]
b)[tex]\:<\:2,-3,4\:>\:=\sqrt{29} \cdot \frac{\:<\:2,-3,4\:>\:}{\sqrt{29} }[/tex]
c) [tex]\:<\:3,6,-2\:>\:=7\cdot \frac{\:<\:3,6,-2\:>\:}{7}[/tex]
Step-by-step explanation:
a) Let a=<2,1,-3>
The magnitude of a is [tex]|a|=\sqrt{2^2+1^2+(-3)^2}[/tex]
[tex]|a|=\sqrt{4+1+9}=\sqrt{14}[/tex]
The unit vector in the direction of a is
[tex]\hat{a}=\frac{\:<\:2,1,-3\:>\:}{\sqrt{14} }[/tex]
Using the relation [tex]a=|a|\hat{a}[/tex], we have
[tex]\:<\:2,1,-3\:>\:=\sqrt{14}\cdot \frac{\:<\:2,1,-3\:>\:}{\sqrt{14} }[/tex]
b) Let a=2i - 3j + 4k
[tex]|a|=\sqrt{2^2+(-3)^2+4^2}[/tex]
[tex]|a|=\sqrt{4+9+16}=\sqrt{29}[/tex]
[tex]\hat{a}=\frac{\:<\:2,-3,4\:>\:}{\sqrt{29} }[/tex]
Using the relation [tex]a=|a|\hat{a}[/tex], we have
[tex]\:<\:2,-3,4\:>\:=\sqrt{29} \cdot \frac{\:<\:2,-3,4\:>\:}{\sqrt{29} }[/tex]
c) Let us first find the sum of <1, 2, -3> and <2, 4, 1> to get:
<1+2, 2+4, -3+1>=<3, 6, -2>
Let a=<3, 6, -2>
The magnitude is
[tex]|a|=\sqrt{3^2+6^2+(-2)^2}[/tex]
[tex]|a|=\sqrt{9+36+4}=\sqrt{49}=7[/tex]
The unit vector in the direction of a is
[tex]\hat{a}=\frac{\:<\:3,6,-2\:>\:}{7}[/tex]
Using the relation [tex]a=|a|\hat{a}[/tex], we have
[tex]\:<\:3,6,-2\:>\:=7\cdot \frac{\:<\:3,6,-2\:>\:}{7}[/tex]
