• anonymous
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
OCW Scholar - Multivariable Calculus
  • Stacey Warren - Expert
Hey! We 've verified this expert answer for you, click below to unlock the details :)
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
  • chestercat
I got my questions answered at in under 10 minutes. Go to now for free help!
  • IrishBoy123
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
  • anonymous
Thank you! I've got it now

Looking for something else?

Not the answer you are looking for? Search for more explanations.