Question

Find two vectors v₁ and v₂ whose sum is (4, 4), where v₁ is parallel to (-3, 4) while v₂ is perpendicular to (-3, 4).

Answer 1

To find two vectors v₁ and v₂ whose sum is (4, 4), we can use the concept of perpendicular components. By setting up equations based on the parallel and perpendicular properties of the given vectors, we can solve for the values of the components of v₁ and v₂. Adding the corresponding components will give us the desired vectors.

Step-by-step explanation:

To find two vectors v₁ and v₂ whose sum is (4, 4), we need to understand that the sum of two vectors is obtained by adding their corresponding components.

Given that v₁ is parallel to (-3, 4) and v₂ is perpendicular to (-3, 4), we can use the concept of perpendicular components to find the values of v₁ and v₂.

Let's assume that v₁ = (a, b) and v₂ = (c, d).

1. Since v₁ is parallel to (-3, 4), the ratio of their corresponding components will be the same. So, -3/a = 4/b. We can solve these equations to find the values of a and b.
2. Since v₂ is perpendicular to (-3, 4), the dot product of v₂ and (-3, 4) will be zero. So, the dot product of (c, d) and (-3, 4) will be zero. Using this equation, we can solve for the values of c and d.
3. Once we have the values of a, b, c, and d, we can add the corresponding components to find the vectors v₁ and v₂.

For example, let's say we find that v₁ = (2, 3) and v₂ = (2, 1), then their sum will be (2 + 2, 3 + 1) = (4, 4).