\(2 \) Given the vectors \( u = ( 4,2 , - 1 ) \) and \( v = ( - 6 , x , - 2 ) \) . i) Find the value(s) of \( x \) if \( u \) and \( v \) are orthogonal vectors. ii) Find a vector with magnitude \( 5 \) in the same direction to vector \( u \) .
Pregunta
Answer
\(i \quad u x v = 0 = 4 x ( - 6 ) + 2 x + 2 = 0\)
\(x = 11\)
ii \( \left. \begin{array} { l }{ 5 = \sqrt { ( 4 x ) ^ { 2 } + ( 2 x ) ^ { 2 } + 1 x ^ { 2 } } }\\{ = \sqrt { 20 x ^ { 2 } } }\\{ x = \sqrt { \frac { 5 } { 4 } } = \frac { \sqrt { 5 } } { 2 } }\\{ ( 2 \sqrt { 5 } , \sqrt { 5 } , - \frac { \sqrt { 5 } } { 2 } ) } \end{array} \right.\)
Related Questions
Q:
Find the inverse for each of the following functions.
\(f ( x ) = 3 x + 1\)
\(g ( x ) = 1 x ^ { 3 } - 5\)
\(h ( x ) = \frac { 3 } { x + 5 }\)
\(j ( x ) = \sqrt[ 3 ] { x + 1 }\)
Q:
The total revenue from the sale of a popular book is approximated by the rational function \( R ( x ) = \frac { 1700 x ^ { 2 } } { x ^ { 2 } + 4 } \) , where \( x \) is the number of years since publication and \( R ( x ) \)
is the total revenue in millions of dollars.
(a) Find the total revenue at the end of the first year.
(b) Find the total revenue at the end of the second year.