A square is defined by the unit vectors i and j. Find the projections of i and j on each of the diagonals of the square.
1/sqrt2 onto each diagonal for both i and j
To find the projections of the unit vectors i and j on each of the diagonals of the square, we first need to determine the equations of the diagonals.
Let's assume that the square has vertices at points A, B, C, and D, where A is at the origin (0, 0) and B is on the x-axis, let's say (a, 0), where "a" is the length of the side of the square.
We can then determine C and D by rotating A and B by 90 degrees. The coordinates of C will be (-a, a), and the coordinates of D will be (0, a).
Now, let's find the equations of the diagonals. The diagonal AC can be defined by the points A and C. The slope of AC can be found by using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points.
m = (a - 0) / (-a - 0)
m = -1
Using the point-slope form of a line (y - y1) = m(x - x1) and substituting the coordinates of point A (0, 0), we get the equation of AC:
y - 0 = -1(x - 0)
y = -x
Similarly, the diagonal BD can be defined by the points B and D. The slope of BD can be found as:
m = (a - 0) / (0 - a)
m = 1
Using the point-slope form of a line and substituting the coordinates of point B (a, 0), we get the equation of BD:
y - 0 = 1(x - a)
y = x - a
Now, to find the projections of the unit vectors i and j on each of these diagonals, we need to determine the components of these vectors along the diagonal lines.
The projection of a vector can be found by taking the dot product of the vector and a unit vector along the direction of the diagonal line. Since the diagonals AC and BD are perpendicular, the unit vectors along their directions are parallel to i and j, respectively.
Therefore, the projections of i on AC and BD are:
Projection of i on AC = i · (-1, -1) = -1
Projection of i on BD = i · (1, 0) = 1
Similarly, the projections of j on AC and BD are:
Projection of j on AC = j · (-1, -1) = -1
Projection of j on BD = j · (0, 1) = 1
Thus, the projections of i on each of the diagonals are -1 and 1, and the projections of j on each of the diagonals are -1 and 1.