Vectors

Revise Vectors unit from G.C.E (A/L) Combined Mathematics.

Direction Cosines

Suppose $\vec{p} = a\underline{i}+b\underline{j}+c\underline{k}$. Direction cosines of $p$ are $\cos{\alpha}, \cos{\beta},\cos{\gamma}$ where $\alpha,\beta,\gamma$ are the angles $p$ makes with $x,y,z$ axes.

Unit vector in the direction of $\vec{p}=\underline{i}\cos{\alpha}+\underline{j}\cos{\beta}+\underline{k}\cos{\gamma}$. Because of this: $$

$$\cos^2{\alpha}+\cos^2{\beta}+\cos^2{\gamma}=1$$

Direction Ratio

Ratio of the direction cosines is called as direction ratio.

$$\cos{\alpha}\,:\,\cos{\beta}\,:\,\cos{\gamma}$$

Cross Product

$$a \times b = \lvert a \rvert \lvert b \rvert sin{(\theta)} n = \begin{vmatrix} \underline{i} & \underline{j} & \underline{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$$

$n$ is the unit normal vector to $a$ and $b$. Direction is based on the right hand rule.

$a \times b = 0 \implies \lvert a \rvert = 0 \lor \lvert b \rvert = 0 \lor a \parallel b$

Cross products between $i$, $j$, $k$ are circular.

Properties

• $a \times a = 0$
• $(a\times b) = -(b \times a)$
• $a \times (b + c) = (a \times b) + (a \times c)$
• Area of a parallelogram $ABCD = \lvert \vec{AB} \times \vec{AD} \rvert$ $$

Scalar Triple Product

$$[a,b,c] = a \cdot (b \times c) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}$$

Properties

• $[a,b,c] = a \cdot (b \times c) = (a \times b) \cdot c$
• $[a,b,c] = [b,c,a] = [c,a,b] = -[a,c,b]$
• Swapping any 2 vectors will negate the product.
• $[a,b,c] = 0$ iff $a$, $b$, $c$ are coplanar.
• Volume of a parallelepiped with $a$, $b$, $c$ as adjacent edges = $[a,b,c]$
• Volume of a tetrahedron with $a$, $b$, $c$ as adjacent edges = $\frac{1}{6}[a,b,c]$

Vector Triple Product

$a \times (b \times c) = (a \cdot c)b - (a \cdot b)c$

Resulting vector lies in the plane that contains $b$ and $c$.