Key Points

The position vector of a point $P(x, y, z)$ with respect to the origin $O(0,0,0)$ is $\vec{r} = \overrightarrow{OP} = x\hat{i} + y\hat{j} + z\hat{k}$. Its magnitude is given by $|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$.

A unit vector has a magnitude of 1. The unit vector in the direction of a given non-zero vector $\vec{a}$ is denoted by $\hat{a}$ and is calculated as $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$.

The vector from point $P_1(x_1, y_1, z_1)$ to point $P_2(x_2, y_2, z_2)$ is $\overrightarrow{P_1P_2} = \overrightarrow{OP_2} - \overrightarrow{OP_1}$. In component form, this is $\overrightarrow{P_1P_2} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$.

The position vector $\vec{r}$ of a point R that divides the line segment joining points P and Q with position vectors $\vec{a}$ and $\vec{b}$ internally in the ratio $m:n$ is $\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$.

For the midpoint of the line segment joining points P and Q with position vectors $\vec{a}$ and $\vec{b}$, the ratio is $1:1$. The position vector of the midpoint R is $\vec{r} = \frac{\vec{a} + \vec{b}}{2}$.

The scalar product of two non-zero vectors $\vec{a}$ and $\vec{b}$ is defined as $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between them. The result is a scalar quantity.

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, their dot product is $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$. Also, $\vec{a} \cdot \vec{a} = |\vec{a}|^2$.

The cosine of the angle $\theta$ between two non-zero vectors $\vec{a}$ and $\vec{b}$ can be found using the formula $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$.

Two non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendicular (orthogonal) to each other if and only if their scalar product is zero, that is $\vec{a} \cdot \vec{b} = 0$.

The projection of a vector $\vec{a}$ on a non-zero vector $\vec{b}$ is a scalar value given by $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$.

The vector product of two non-zero vectors $\vec{a}$ and $\vec{b}$ is a vector defined as $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$. Here, $\theta$ is the angle between them and $\hat{n}$ is a unit vector perpendicular to the plane of $\vec{a}$ and $\vec{b}$.

For $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, the cross product is calculated using the determinant: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$.

Two non-zero vectors $\vec{a}$ and $\vec{b}$ are collinear or parallel if and only if their vector product is the zero vector, that is $\vec{a} \times \vec{b} = \overrightarrow{0}$. This is equivalent to $\vec{b} = \lambda\vec{a}$ for some scalar $\lambda$.

The area of a parallelogram with adjacent sides represented by vectors $\vec{a}$ and $\vec{b}$ is given by the magnitude of their cross product, Area $= |\vec{a} \times \vec{b}|$.

The area of a triangle with adjacent sides represented by vectors $\vec{a}$ and $\vec{b}$ is half the magnitude of their cross product, Area $= \frac{1}{2}|\vec{a} \times \vec{b}|$.

For a vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, the direction ratios are $a=x, b=y, c=z$. The direction cosines are $l = \frac{x}{|\vec{r}|}, m = \frac{y}{|\vec{r}|}, n = \frac{z}{|\vec{r}|}$, and they satisfy the relation $l^2 + m^2 + n^2 = 1$.