Resolve u=[3,4,7] into two orthogonal vectors, one of which is collinear with v=[1,2,3].

To resolve vector u = [3, 4, 7] into two orthogonal vectors, one of which is collinear with vector v = [1, 2, 3], we can use the concept of vector projection.

First, let's find the scalar projection of u onto v. The scalar projection is given by the formula:

Scalar projection of u onto v = (u . v) / ||v||,

where u . v represents the dot product of u and v, and ||v|| represents the magnitude of vector v.

Calculating the scalar projection:

u . v = (3 * 1) + (4 * 2) + (7 * 3) = 3 + 8 + 21 = 32,

||v|| = sqrt((1^2) + (2^2) + (3^2)) = sqrt(1 + 4 + 9) = sqrt(14).

Scalar projection of u onto v = (32) / sqrt(14) = (32 / sqrt(14)).

Next, we need to find the projection vector. The projection vector is given by:

Projection of u onto v = (scalar projection of u onto v) * (unit vector of v).

To find the unit vector of v, we divide vector v by its magnitude:

Unit vector of v = v / ||v|| = [1, 2, 3] / sqrt(14).

Calculating the projection vector:

Projection of u onto v = (32 / sqrt(14)) * [1, 2, 3] / sqrt(14) = (32 / 14) * [1, 2, 3] = [16/7, 32/7, 48/7].

Finally, to obtain the orthogonal vector, we subtract the projection vector from the original vector u:

Orthogonal vector = u - Projection of u onto v = [3, 4, 7] - [16/7, 32/7, 48/7].

Simplifying the subtraction:

Orthogonal vector = [3 - (16/7), 4 - (32/7), 7 - (48/7)] = [11/7, 12/7, 1/7].

Therefore, the two orthogonal vectors are:
1. The projection vector collinear with v: [16/7, 32/7, 48/7].
2. The orthogonal vector: [11/7, 12/7, 1/7].