Mathematics

Mathematics

Problem

5. Show that each of the given three vectors is a unit vector: 17(2i^+3j^+6k^),17(3i^−6j^+2k^),17(6i^+2j^−3k^)\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \frac{1}{7}(3 \hat{i}-6 \hat{j}+2 \hat{k}), \quad \frac{1}{7}(6 \hat{i}+2 \hat{j}-3 \hat{k}) Also, show that they are mutually perpendicular to each other.
Intuitive explanation

A unit vector has magnitude exactly 1—so your job is to compute the length of each vector using the magnitude formula ∣v⃗∣=vx2+vy2+vz2|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} and verify it equals 1. Then, two vectors are perpendicular when their dot product is zero, so you'll take each pair and show their dot product vanishes. This is a nice check that these three vectors form an orthonormal basis.

Intuitive explanation

A unit vector has magnitude exactly 1—so your job is to compute the length of each vector using the magnitude formula ∣v⃗∣=vx2+vy2+vz2|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} and verify it equals 1. Then, two vectors are perpendicular when their dot product is zero, so you'll take each pair and show their dot product vanishes. This is a nice check that these three vectors form an orthonormal basis.

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