58.2k views
4 votes
Given ⃗ = [3,4,3] and ⃗⃗ = [1, −2,5] a. Calculate the angle between ⃗ and ⃗⃗.

b. If c⃗ = [1, −3,], find so that ⃗ and c⃗ are perpendicular

1 Answer

3 votes

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].

User RayOldProf
by
7.2k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories