Did you know?
Did you know that a vector can be represented by its initial and terminal points? For example, the vector AB has an initial point A=(-7,6) and a terminal point B=(5,1). By using this notation, we can perform various calculations and learn interesting properties about vectors.
1) One way to represent a vector is by its component form. The component form of AB can be found by subtracting the x and y coordinates of A from the x and y coordinates of B. In this case, the component form of AB is [5-(-7), 1-6], which simplifies to [12, -5].
2) Vectors can also be represented as a linear combination of the unit vectors i and j. To write AB as a linear combination, we simply multiply the x-component of AB by i and the y-component by j. So AB can be written as 12i - 5j.
3) The magnitude of a vector represents its length. To find the magnitude of AB, we can use the Pythagorean theorem. The magnitude of AB is equal to the square root of the sum of the squares of its components. In this case, the magnitude of AB is √(12^2 + (-5)^2), which simplifies to √(144 + 25), and further simplifies to √169, which is equal to 13.
4) Unit vectors have a magnitude of 1 and represent the direction of a vector. To find a unit vector with the same direction as AB, we divide AB by its magnitude. Here, the unit vector u can be found by dividing AB by 13, resulting in u = (12/13)i - (5/13)j. The magnitude of u is equal to 1.
5) The direction angle of a vector is the angle it makes with the positive x-axis. To find the direction angle of AB, we can use trigonometry. The direction angle θ can be found by taking the inverse tangent of the y-component divided by the x-component. In this case, the direction angle of AB is tan^(-1)(-5/12), which is approximately -0.39 radians or -22.4 degrees.
