Final answer:
To find the value of k for which the vectors are orthogonal, calculate the dot product of the two vectors and set it equal to zero. The value of k is 1.0.
Step-by-step explanation:
In order for two vectors to be orthogonal, their dot product must be equal to zero. So to find the value of k for which the vectors are orthogonal, we need to calculate the dot product of the two vectors and set it equal to zero.
The dot product of two vectors is given by the formula:
A · B = AxBx + AyBy + AzBz
In this case, the two vectors are:
A = (2.0î - 4.0ĵ + Â)N
B = (3.0î + 4.0ĵ + 10.0Â)N
Substituting the values into the dot product formula, we get:
(2.0)(3.0) + (-4.0)(4.0) + (Â)(10.0) = 0
Simplifying this equation, we have:
6.0 - 16.0 + 10.0Â = 0
Combining like terms, we get:
-10.0 + 10.0Â = 0
Adding 10.0 to both sides:
10.0Â = 10.0
Dividing both sides by 10.0, we find:
 = 1.0
So, the value of k for which the vectors are orthogonal is 1.0.