FOR THE POINT A(1,5,-7) AND B(3,0,1) VECTOR AB HAS COORDINATES
To find the vector AB, we can subtract the coordinates of point A from the coordinates of point B. This can be written as:
AB = B - A
We can then perform the subtraction of the individual coordinates:
AB = (3, 0, 1) - (1, 5, -7)
AB = (2, -5, 8)
Therefore, the vector AB has coordinates (2, -5, 8).
2 i - 5 j + 8 k
The given expression, "2i - 5j + 8k" represents a vector in 3D Cartesian coordinate system.
The coefficients of i, j, and k represent the magnitude of the vector in the x, y, and z directions respectively.
So, the vector can be represented as:
V = 2i - 5j + 8k
This means that the vector has a magnitude of √(2^2 + (-5)^2 + 8^2) = √(4 + 25 + 64) = √93.
The direction of the vector can be determined by calculating the angles that it makes with the positive x, y, and z-axes.
The angle θ with the positive x-axis can be calculated as:
θx = cos^-1(2/√93)
Similarly, the angles θy and θz with the positive y and z-axes can be calculated as:
θy = cos^-1(-5/√93)
θz = cos^-1(8/√93)
Therefore, the vector 2i - 5j + 8k has a magnitude of √93 and makes angles of approximately 16.7°, 163.1°, and 82.6° with the positive x, y, and z-axes respectively.
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To find the coordinates of the vector AB, we need to calculate the differences between the corresponding coordinates of points A and B.
The coordinates of vector AB can be found using the following formulas:
x-component: BX - AX
y-component: BY - AY
z-component: BZ - AZ
Where AX, AY, AZ are the coordinates of point A, and BX, BY, BZ are the coordinates of point B.
Given the coordinates of point A (1, 5, -7) and point B (3, 0, 1), we can substitute these values into the formula to find the coordinates of vector AB:
x-component of AB: 3 - 1 = 2
y-component of AB: 0 - 5 = -5
z-component of AB: 1 - (-7) = 8
Therefore, the coordinates of vector AB are (2, -5, 8).