Find the inner product of a and b if a= (3, 0, –1) and b= (4, –2, 5), and state whether the vectors are perpendicular.
To find the inner product (also known as the dot product) of two vectors, you need to multiply their corresponding components and then sum them up.
Given: a = (3, 0, -1) and b = (4, -2, 5).
To find the inner product of a and b, you can use the following formula:
a · b = (a₁ * b₁) + (a₂ * b₂) + (a₃ * b₃)
Let's calculate it:
a · b = (3 * 4) + (0 * -2) + (-1 * 5)
= 12 + 0 + (-5)
= 7
Therefore, the inner product (a · b) is 7.
Now, to determine if the vectors are perpendicular, you can use the fact that two vectors are perpendicular (orthogonal) if and only if their inner product is zero.
Since the inner product of a and b is 7 (not zero), we can conclude that the vectors a and b are not perpendicular to each other.