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MATH / Let a, b, and c be unit vectors, such that the angle between a and b is arccos(1)/(5), the angle between a and c is arccos(1)/(6), and the angle between b and c is 60^(∘). Let P be…

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problem

Let a,\mathbf{a}, b,\mathbf{b}, and c\mathbf{c} be unit vectors, such that the angle between a\mathbf{a} and b\mathbf{b} is arccos15,\arccos \frac{1}{5}, the angle between a\mathbf{a} and c\mathbf{c} is arccos16,\arccos \frac{1}{6}, and the angle between b\mathbf{b} and c\mathbf{c} is 60∘.60^\circ. Let PP be the plane which contains the origin, b,\mathbf{b}, and c.\mathbf{c}. Then the projection of a\mathbf{a} onto PP can be expressed in the form pb+qcp \mathbf{b} + q \mathbf{c}for some scalars pp and q.q. Enter the ordered pair (p,q).(p,q).
Plain-text mathematical notation (without MathML)
Let a, b, and c be unit vectors, such that the angle between a and b is arccos(1)/(5), the angle between a and c is arccos(1)/(6), and the angle between b and c is 60^(∘).

Let P be the plane which contains the origin, b, and c.  Then the projection of a onto P can be expressed in the form
pb+qcfor some scalars p and q.  Enter the ordered pair (p,q).
Original LaTeX notation
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors, such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\arccos \frac{1}{5},$ the angle between $\mathbf{a}$ and $\mathbf{c}$ is $\arccos \frac{1}{6},$ and the angle between $\mathbf{b}$ and $\mathbf{c}$ is $60^\circ.$

Let $P$ be the plane which contains the origin, $\mathbf{b},$ and $\mathbf{c}.$  Then the projection of $\mathbf{a}$ onto $P$ can be expressed in the form
\[p \mathbf{b} + q \mathbf{c}\]for some scalars $p$ and $q.$  Enter the ordered pair $(p,q).$

level

Level 5

type

Precalculus

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task_id
29e3618b-58ef-565a-b36d-61f70c5464a2
task_key
precalculus--test--29e3618b-58ef-565a-b36d-61f70c5464a2
task_revision_id
1
config / split
precalculus / test

Official source

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