Question

Let v = (v1, v2) be a vector in r2. show that (v2, −v1) is orthogonal to v, and use this fact to find two unit vectors orthogonal to the given vector. v = (9, 40)

Answer 1

To show that (v2, -v1) is orthogonal to v, take the dot product of the two vectors and show that it equals zero. To find two unit vectors orthogonal to v, swap the components of v and change the sign of one of them, then divide the resulting vector by its magnitude to get a unit vector.

Step-by-step explanation:

To show that (v2, -v1) is orthogonal to v, we can take the dot product of the two vectors and show that it equals zero. The dot product of two vectors is calculated by multiplying their corresponding components and then summing them up. For the given vector v = (9, 40) and (v2, -v1) = (40, -9), the dot product is: 9 * 40 + 40 * -9 = 0.

To find two unit vectors orthogonal to v, we can use the fact that any multiple of an orthogonal vector is still orthogonal. First, we find one orthogonal vector by swapping the components of v and changing the sign of one of them: (v2, -v1) = (40, -9). Then, we can divide this vector by its magnitude to get a unit vector: (40 / sqrt(40^2 + (-9)^2), -9 / sqrt(40^2 + (-9)^2)) = (40 / sqrt(1691), -9 / sqrt(1691)). Finally, we can multiply this unit vector by any scalar value to get two different unit vectors orthogonal to v.