Final answer:
To find two unit vectors orthogonal to both vector1 and vector2, we can use the cross product of the two vectors. Divide the resulting cross product by its magnitude to get a unit vector orthogonal to both vector1 and vector2.
Step-by-step explanation:
To find two unit vectors orthogonal to both vector1 and vector2, we can use the cross product of the two vectors. Let vector1 = a and vector2 = b. The cross product of a and b is given by the formula: a x b = |a||b|sin(theta)n, where theta is the angle between the two vectors and n is a unit vector perpendicular to the plane formed by the two vectors.
Since we want two unit vectors, we can divide the resulting cross product by its magnitude to get a unit vector orthogonal to both vector1 and vector2.
Let's explain with an example: Suppose vector1 = i + 2j + 3k and vector2 = 2i + j + k. To find two unit vectors orthogonal to both vector1 and vector2, we can calculate:
a x b = (i + 2j + 3k) x (2i + j + k) = (2i + 4j + 6k) x (2i + j + k) = 2(i x i) + 4(j x i) + 6(k x i) + 2(i x j) + 4(j x j) + 6(k x j) + 2(i x k) + 4(j x k) + 6(k x k) = -2j + 4k - 2i + 6k - 4i - 2j + 12i - 6j = 6i - 8j + 10k
The magnitude of a x b is |a x b| = sqrt((6^2) + (-8^2) + (10^2)) = sqrt(100) = 10. Dividing the cross product by its magnitude, we get the unit vector:
n = (6i - 8j + 10k)/10 = (3/5)i - (4/5)j + (1/2)k
So, the two unit vectors orthogonal to both vector1 and vector2 are (3/5)i - (4/5)j + (1/2)k and -(3/5)i + (4/5)j - (1/2)k.