From: Andrew Lorimer Date: Tue, 29 Jan 2019 22:46:37 +0000 (+1100) Subject: vector projections and axes X-Git-Tag: yr12~274 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/d9867284690d15354ce468f884f2e0ceee62ad60?hp=b7b81d96f2b4cc44a0ceafd493a707ad8a0b1f15 vector projections and axes --- diff --git a/spec/vectors.md b/spec/vectors.md index 6218b24..0a95e9e 100644 --- a/spec/vectors.md +++ b/spec/vectors.md @@ -152,3 +152,15 @@ $$\therefore \vec{c} = m\vec{a} + n\vec{b} \quad \text{(simultaneous)}$$ $\vec{a}, \vec{b},$ and $\vec{c}$ are linearly independent if no vector in the set is expressible as a linear combination of other vectors in set, or if they are parallel. Vector $\vec{w}$ is a linear combination of vectors $\vec{v_1}, \vec{v_2}, \vec{v_3}$ + +## Three-dimensional vectors + +Right-hand rule for axes - $z$ is up or out of page. + +## Angle between vector and axis + +Direction of a vector can be given by the angles it makes with $\vec{i}, \vec{j}, \vec{k}$ directions. + +For $\vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes: +$$\cos \alpha = {a_1 \over |\vec{a}|}, \quad \cos \beta = {a_2 \over |\vec{a}|}, \quad \cos \gamma = {a_3 \over |\vec{a}|}$$ +