Given [tex]$p=3 i+2 k, q=4 i-2 j+3 k$[/tex], determine [tex]$2 p+3 q$[/tex]
Answer (1)
Multiply vector p by 2: 2 p = 6 i + 4 k .
Multiply vector q by 3: 3 q = 12 i − 6 j + 9 k .
Add the resulting vectors: 2 p + 3 q = ( 6 i + 4 k ) + ( 12 i − 6 j + 9 k ) .
Simplify to get the final answer: 18 i − 6 j + 13 k .
Explanation
Understanding the Problem We are given two vectors, p = 3 i + 2 k and q = 4 i − 2 j + 3 k . Our objective is to find the vector 2 p + 3 q . This involves scalar multiplication of vectors and vector addition.
Multiplying p by 2 First, we multiply the vector p by the scalar 2: 2 p = 2 ( 3 i + 2 k ) = 2 ( 3 i ) + 2 ( 2 k ) = 6 i + 4 k
Multiplying q by 3 Next, we multiply the vector q by the scalar 3: 3 q = 3 ( 4 i − 2 j + 3 k ) = 3 ( 4 i ) + 3 ( − 2 j ) + 3 ( 3 k ) = 12 i − 6 j + 9 k
Adding the Vectors Now, we add the resulting vectors 2 p and 3 q :
2 p + 3 q = ( 6 i + 4 k ) + ( 12 i − 6 j + 9 k ) = ( 6 i + 12 i ) + ( − 6 j ) + ( 4 k + 9 k ) 2 p + 3 q = ( 6 + 12 ) i − 6 j + ( 4 + 9 ) k = 18 i − 6 j + 13 k
Final Answer Therefore, the vector 2 p + 3 q is 18 i − 6 j + 13 k .
Examples
Vector operations are used extensively in physics and engineering. For example, when analyzing forces acting on an object, each force can be represented as a vector. If you have two forces acting on an object, say F 1 = 3 i + 2 k and F 2 = 4 i − 2 j + 3 k , the net force is the sum of these vectors. If you need to double the first force and triple the second force, you would calculate 2 F 1 + 3 F 2 in the same way we calculated 2 p + 3 q in this problem. This helps determine the overall effect on the object's motion.
