Let $a =2 i +9 j , b =-4 i +5 j , c = i -3 j$, and $d =14 j$ where $i$ and $j$ are unit vectors.
What is $( a - b )-(2 d - c )$ in terms of $i$ and $j$?
A. $5 i-27 j$
B. $5 i+35 j$
C. $7 i -27 j$
D. $7 i+39 j
Answer (1)
Calculate a − b = ( 2 i + 9 j ) − ( − 4 i + 5 j ) = 6 i + 4 j .
Calculate 2 d = 2 ( 14 j ) = 28 j .
Calculate 2 d − c = 28 j − ( i − 3 j ) = − i + 31 j .
Calculate ( a − b ) − ( 2 d − c ) = ( 6 i + 4 j ) − ( − i + 31 j ) = 7 i − 27 j .
The final answer is 7 i − 27 j .
Explanation
Problem Setup We are given vectors a = 2 i + 9 j , b = − 4 i + 5 j , c = i − 3 j , and d = 14 j . Our goal is to find the vector ( a − b ) − ( 2 d − c ) in terms of i and j .
Calculating a - b First, we calculate a − b :
a − b = ( 2 i + 9 j ) − ( − 4 i + 5 j ) = ( 2 − ( − 4 )) i + ( 9 − 5 ) j = ( 2 + 4 ) i + 4 j = 6 i + 4 j
Calculating 2d Next, we calculate 2 d :
2 d = 2 ( 14 j ) = 28 j
Calculating 2d - c Now, we calculate 2 d − c :
2 d − c = ( 28 j ) − ( i − 3 j ) = − i + ( 28 − ( − 3 )) j = − i + ( 28 + 3 ) j = − i + 31 j
Calculating (a - b) - (2d - c) Finally, we calculate ( a − b ) − ( 2 d − c ) :
( a − b ) − ( 2 d − c ) = ( 6 i + 4 j ) − ( − i + 31 j ) = ( 6 − ( − 1 )) i + ( 4 − 31 ) j = ( 6 + 1 ) i + ( 4 − 31 ) j = 7 i − 27 j
Final Answer Therefore, ( a − b ) − ( 2 d − c ) = 7 i − 27 j .
Examples
Vector operations are used in physics to calculate resultant forces, velocities, and accelerations. For example, if two forces are acting on an object, we can represent them as vectors and add them to find the net force. Similarly, in computer graphics, vectors are used to represent positions, directions, and transformations of objects in 3D space. Understanding vector operations is crucial for simulations, game development, and various engineering applications.
