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 v=[1,2,3], we can use the projection formula.
The formula for the projection P of a vector u onto another vector v is given by:
P = (u · v) / ||v||^2 * v
where · represents the dot product and ||v|| represents the magnitude of vector v.
First, we calculate the dot product of u and v:
u · v = 3 * 1 + 4 * 2 + 7 * 3 = 3 + 8 + 21 = 32
Next, we calculate the magnitude of vector v:
||v|| = √(1^2 + 2^2 + 3^2) = √(1 + 4 + 9) = √14
Now, we substitute the values into the projection formula:
P = (32 / 14) * [1,2,3] = (16/7) * [1,2,3]
This resulting vector P is collinear with v and can be considered as one of the orthogonal vectors we need.
To find the second orthogonal vector, we subtract P from u:
Orthogonal vector A = u - P = [3,4,7] - (16/7) * [1,2,3]
Element-wise subtraction gives:
A = [3,4,7] - [(16/7) * 1, (16/7) * 2, (16/7) * 3]
Simplifying further:
A = [3,4,7] - [16/7, 32/7, 48/7]
Element-wise subtraction gives:
A = [3 - 16/7, 4 - 32/7, 7 - 48/7]
Calculating the subtraction:
A = [(21 - 16) / 7, (28 - 32) / 7, (49 - 48) / 7]
Simplifying the difference:
A = [5/7, -4/7, 1/7]
Therefore, the two orthogonal vectors are:
P = (16/7) * [1,2,3]
A = [5/7, -4/7, 1/7]
And both vectors are orthogonal to each other, with vector P being collinear with vector v=[1,2,3].