Give me the explanation of Rewrite a vector in component form; and
Answer (1)
To rewrite a vector in component form means to express the vector as the sum of its components along each axis of a coordinate system. Let's break it down:1. Understanding Vectors:A vector is a quantity that has both magnitude (size or length) and direction. It's often represented graphically as an arrow. In a two-dimensional (2D) space (like a flat plane), a vector has two components: one along the x-axis (horizontal) and one along the y-axis (vertical). In three-dimensional (3D) space, it has three components: x, y, and z.2. Component Form Representation:The component form of a vector is written as an ordered pair (in 2D) or an ordered triple (in 3D), enclosed in angle brackets or parentheses. Each number in the pair or triple represents the vector's component along a specific axis.2D: A vector v with an x-component of 'a' and a y-component of 'b' is written as: ⟨a, b⟩ or (a, b)3D: A vector v with x-component 'a', y-component 'b', and z-component 'c' is written as: ⟨a, b, c⟩ or (a, b, c)3. How to Rewrite a Vector in Component Form:The method depends on how the vector is initially given:If given graphically (as an arrow):Choose a coordinate system.Determine the vector's x-component by measuring the horizontal distance from the tail to the head of the arrow. A positive value indicates movement to the right, and a negative value indicates movement to the left.Determine the vector's y-component by measuring the vertical distance from the tail to the head of the arrow. A positive value indicates movement upward, and a negative value indicates movement downward.For 3D vectors, you'll similarly measure the z-component.Write the vector in component form using the measured values.If given using magnitude and direction (polar form):Use trigonometry (sine and cosine functions). The x-component is the magnitude multiplied by the cosine of the direction angle, and the y-component is the magnitude multiplied by the sine of the direction angle. For 3D vectors, you'll need to use more complex trigonometric functions.If given as a difference of two points:Subtract the coordinates of the tail point from the coordinates of the head point. The result is the component form of the vector.Example (2D):Let's say a vector is represented graphically, and by measuring, we find its horizontal component is 3 units to the right, and its vertical component is 4 units upward. Then, the component form of the vector is ⟨3, 4⟩.In summary, rewriting a vector in component form is a way to represent the vector numerically, making it easier to perform mathematical operations on it. The specific method used depends on how the vector is initially presented.
