How To Write Component Form Of Vectors

The component form of a vector is a way of expressing a vector as the sum of its components along coordinate axes. In a three-dimensional Cartesian coordinate system, a vector (\mathbf{v}) can be represented as:

[\mathbf{v} = \langle v1, v2, v_3 \rangle]

where (v1, v2, v_3) are the components of the vector along the x, y, and z axes, respectively.

To find the component form of a vector, you need its initial and terminal points in space. Let's say the initial point is ((x1, y1, z1)) and the terminal point is ((x2, y2, z2)). Then, the components can be found by subtracting the corresponding coordinates:

[v1 = x2 - x_1]

[v2 = y2 - y_1]

[v3 = z2 - z_1]

For example, let's say you have two points A((x1, y1, z1)) and B((x2, y2, z2)), and you want to find the vector (\overrightarrow{AB}). The component form would be:

[\overrightarrow{AB} = \langle x2 - x1, y2 - y1, z2 - z1 \rangle]

Here are the general steps:

1. Identify the initial point ((x1, y1, z1)) and terminal point ((x2, y2, z2)) of the vector.
2. Subtract corresponding coordinates to find the components (v1, v2, v_3).
3. Write the vector in component form as (\langle v1, v2, v_3 \rangle).

Keep in mind that the order of subtraction matters. If you switch the order of the points, the components will have the opposite sign. The direction of the vector is determined by the order of subtraction.

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