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Vector subtraction is a fundamental operation used to find the difference between two vectors. It involves subtracting the corresponding components of the vectors. This process is crucial to determine the vector that lies between two points, say Point A and Point B. To subtract vector coordinates, begin by taking each corresponding component of the vectors and subtract them starting from the initial point to the terminal point. For instance, when finding

• \overrightarrow{AB}

subtract the coordinates of point A from B: \[B - A = (-5, -3, 1) - (5, 3, -1) = (-10, -6, 2) \] This operation gives the change in each spatial dimension from point A to point B, hence producing vector

• \overrightarrow{AB}: (-10, -6, 2)

Repeat the same process for \overrightarrow{BC} and \overrightarrow{AC} to find the differences between other sets of points. Such calculations lay the groundwork for subsequent magnitude computations and analyses of vector relationships, like determining collinearity.

The magnitude of a vector is a measure of its length, calculated using the formula: \[\|\overrightarrow{v}\| = \sqrt{x^2 + y^2 + z^2} \] where \overrightarrow{v} is represented by its components (x, y, z). This formula comes from applying the Pythagorean theorem in three-dimensional space, providing the straight-line distance of the vector.Taking our previous vector,

• \overrightarrow{AB} = (-10, -6, 2),

we can compute its magnitude as \[\|\overrightarrow{AB}\| = \sqrt{(-10)^2 + (-6)^2 + (2)^2} = \sqrt{140}.\] The calculation gives us the scalar length of the vector. Similarly, the magnitude of \overrightarrow{BC} is calculated the same way because its components are identical to \overrightarrow{AB}, and both result in

• \sqrt{140}.

For \overrightarrow{AC}, however, the magnitude is \[\|\overrightarrow{AC}\| = \sqrt{(-20)^2 + (-12)^2 + (4)^2} = \sqrt{560}.\] This lengthier calculation reflects on the extended length of the vector AC due to larger changes in spatial components.

Collinearity is a specific condition that holds when three or more points lie on the same straight line. This property can be checked by verifying if the sum of magnitudes of two vectors equals the magnitude of the combined vector extending over the three points. Our given problem explores the points A, B, and C. For these points to be collinear, the equation \[\|\overrightarrow{AB}\| + \|\overrightarrow{BC}\| = \|\overrightarrow{AC}\| \] must be satisfied.Utilizing the computed magnitudes:

• \|\overrightarrow{AB}\| = \sqrt{140}
• \|\overrightarrow{BC}\| = \sqrt{140}
• \|\overrightarrow{AC}\| = \sqrt{560}

Check the condition: \[\sqrt{140} + \sqrt{140} = \sqrt{280},\] which is equal to \[\sqrt{560},\]meaning both sides are equivalent. Therefore, the collinearity condition is satisfied, confirming that points A, B, and C are indeed collinear.