Question

The direction of a space vector is often given by its direction cosines. To describe these, let A = a i + b j + c k be a space vector,
represented as an origin vector, and let α, β, and γ be the three angles (≤ π) that A makes
respectively with i , j, and k .
a) Show that dir A = cos α i + cos β j + cos γ k . (The three coefficients are
called the direction cosines of A.)
b) Express the direction cosines of A in terms of a, b, c; find the direction cosines
of the vector − i + 2 j + 2 k .
c) Prove that three numbers t, u, v are the direction cosine

Answer 1

a) by definition: \(dir \ \vec A = \frac{\vec A}{|\vec A| } \ [[ \ = \hat A]] \ !!!\) where
\(| \vec A| = \sqrt{a^2 + b^2 + c^2}\) creating \(dir \ \vec A = \frac{a \hat i + b \hat j + c \hat k}{\sqrt{a^2 + b^2 + c^2} }\)

so you can project vectors using the dot product.

thus \( dir \ \vec A \bullet \hat i \), which projects \( \frac{ \vec A}{|\vec A|} \ [[ \ = \hat A]] \) onto the *unit* \( \hat i\) vector, gives, \(= \frac{a}{ \sqrt{a^2 + b^2 + c^2}} \) which \(= \frac{\vec A}{|\vec A| } | \hat i| \ cos \alpha = (1 \times 1) \ cos \ alpha = \ cos \alpha \)

and the same for b and c

b) it follows that: \(cos \alpha = \frac{a}{ \sqrt{a^2 + b^2 + c^2}} \),
\(cos \beta = \frac{b}{ \sqrt{a^2 + b^2 + c^2}} \),
etc

and the numerical follows from this too

c) prove that 3 numbers as opposed to 3 cows or 3 pizzas? maybe we need some more colour on this one. not sure i understand the question

Answer 2

Thank you! I've got it now