Consider the inner product space R⁴ with respect to the standard inner product. If the vectors u = (3, 2, k, -5) and v = (1, k, 7, 3) are orthogonal, find the value of k.
Answer (1)
To find the value of k for which the vectors u = ( 3 , 2 , k , − 5 ) and v = ( 1 , k , 7 , 3 ) are orthogonal in the inner product space R 4 , we use the concept of the dot product.
The dot product of two vectors a = ( a 1 , a 2 , a 3 , a 4 ) and b = ( b 1 , b 2 , b 3 , b 4 ) in R 4 is given by:
a ⋅ b = a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 .
For the vectors to be orthogonal, their dot product must be equal to 0.
Applying this to our vectors u and v :
u ⋅ v = ( 3 ) ( 1 ) + ( 2 ) ( k ) + ( k ) ( 7 ) + ( − 5 ) ( 3 ) .
Simplifying, we get:
3 + 2 k + 7 k − 15 = 0.
Combine the k terms and the constants:
9 k − 12 = 0.
Add 12 to both sides:
9 k = 12.
Finally, divide by 9:
k = 9 12 = 3 4 .
Therefore, the value of k that makes the vectors u and v orthogonal is 3 4 .
