How can the magnitude of a vector in three-space be determined? (I:1)

Question

BlackTom AI · Accepted Answer

Consider the question: How can the magnitude of a vector in three-space be determined?

Option a: 'By dividing the components by their sum' — This operation does not yield a magnitude; dividing components by their sum would mix components without producing a length, and there is no standard vector magnitude formula that uses such a ratio. It’s a misapplication of normalization concepts without squaring and summing.

Option b: 'By taking the dot product with itself and then square rooting the result' — This aligns with the standard definition of the Euclidean magnitude. If the vector is v = (x, y, z), then v · v = x^2 + y^2 + z^2, and the magnitude is sqrt(x^2 + y^2 + z^2). This is the correct, conventional method for three-dimensional vectors.

Option c: 'By calculating the area under the curve' — The magnitude of a vector is not related to an area under a curve; this phrase belongs to calculus contexts like integrals or velocity–time graphs, not to vector length computations.

Option d: 'By adding the components' — Simply summing x, y, and z does not produce a magnitude; the magnitude involves squaring components, summing, and taking a square root, not a straightforward sum of components.

In summary, the only approach that faithfully computes the Euclidean length of a 3D vector is the squared-sum-and-root method described in option b, while the other options rely on incorrect operations for determining magnitude.

How can the magnitude of a vector in three-space be determined? (I:1)

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