Answer:
a. To calculate the angle between two vectors, you can use the dot product formula and the cosine formula.
Explanation:
The dot product of two vectors ⃗ and ⃗⃗ is given by:
⃗ ⋅ ⃗⃗ = |⃗| * |⃗⃗| * cos(θ),
where |⃗| represents the magnitude of vector ⃗, |⃗⃗| represents the magnitude of vector ⃗⃗, and θ is the angle between them.
Given ⃗ = [3, 4, 3] and ⃗⃗ = [1, -2, 5], their magnitudes are:
|⃗| = √(3² + 4² + 3²) = √34,
|⃗⃗| = √(1² + (-2)² + 5²) = √30.
The dot product ⃗ ⋅ ⃗⃗ is:
⃗ ⋅ ⃗⃗ = 3 * 1 + 4 * (-2) + 3 * 5 = 3 - 8 + 15 = 10.
Now you can use the formula to find the cosine of the angle θ:
cos(θ) = (⃗ ⋅ ⃗⃗) / (|⃗| * |⃗⃗|) = 10 / (√34 * √30).
Then, the angle θ is:
θ = arccos(cos(θ)).
b. Two vectors are perpendicular if their dot product is zero. Given ⃗ = [3, 4, 3] and c⃗ = [1, -3], we can add a third component of 0 to c⃗ to match the dimensionality of ⃗. So, c⃗ becomes [1, -3, 0].
For ⃗ and c⃗ to be perpendicular, their dot product must be zero:
⃗ ⋅ c⃗ = 3 * 1 + 4 * (-3) + 3 * 0 = 3 - 12 + 0 = -9.
So, the value of the missing component in c⃗ is -9:
c⃗ = [1, -3, -9].