Vector Algebra

1Vectors, addition and components

A vector has magnitude and direction; vectors add by the triangle or parallelogram law. A unit vector is \hat{a}=\dfrac{\vec{a}}{|\vec{a}|}, and any vector in space is \vec{a}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k} with magnitude \sqrt{a_x^2+a_y^2+a_z^2}.

Figure. Parallelogram law in the plane: \vec{a} and \vec{b} from a common origin, diagonal \vec{a}+\vec{b}. Not to scale for any particular magnitude.

Reading a component form

1. List the componentsWrite \vec{a}=(a_x,a_y,a_z) along \hat{i},\hat{j},\hat{k}.
2. Magnitude|\vec{a}|=\sqrt{a_x^2+a_y^2+a_z^2}.
3. Unit vectorDivide by the magnitude: \hat{a}=\vec{a}/|\vec{a}|.

2Dot product and angle

The dot product is \vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta=a_xb_x+a_yb_y+a_zb_z. It is zero for perpendicular vectors. Perpendicularity is instant from a vanishing dot product — pick that test when the question hints at a right angle.

Figure. Dot product pairs magnitude with the cosine of the included angle; it vanishes exactly when the vectors are perpendicular.

Finding the angle

1. Dot the componentsSum a_xb_x+a_yb_y+a_zb_z.
2. Both magnitudesCompute |\vec{a}| and |\vec{b}|.
3. Solve for \theta\cos\theta=(\vec{a}\cdot\vec{b})/(|\vec{a}||\vec{b}|).

3Projection of one vector on another

The scalar projection of \vec{a} on \vec{b} is \dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}. It is the signed length of the shadow of \vec{a} along the direction of \vec{b} — positive when the angle is acute, negative when obtuse.

Figure. The projection of \vec{a} on \vec{b} is the signed segment along \vec{b} cut by the perpendicular from the tip of \vec{a}.

4Cross product and area

The cross product \vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n} is a vector perpendicular to both, with magnitude equal to the area of the parallelogram they span. It is zero for parallel vectors and anti-commutative. Parallelism is instant from a zero cross product.

Figure. Right triangle in the coordinate plane with legs 1 and 2 (to scale along each axis separately). Area \tfrac12\cdot 1\cdot 2=1 matches \tfrac12|\vec{AB}\times\vec{AC}|.

Triangle area from two edges

1. Form two edge vectorsFrom a common vertex, \overrightarrow{AB} and \overrightarrow{AC}.
2. Cross themCompute \overrightarrow{AB}\times\overrightarrow{AC} (determinant form in components).
3. Halve the magnitudeArea =\tfrac12|\overrightarrow{AB}\times\overrightarrow{AC}|.

5Scalar triple product and coplanarity

The scalar triple product [\vec{a}\ \vec{b}\ \vec{c}]=\vec{a}\cdot(\vec{b}\times\vec{c}) equals the volume of the parallelepiped formed by the three vectors. It is zero when the vectors are coplanar — so coplanarity is the vanishing of that triple product.

Testing coplanarity

1. Form the determinantEvaluate \vec{a}\cdot(\vec{b}\times\vec{c}) as a 3\times 3 determinant of components.
2. Read the signZero means coplanar; non-zero means a genuine parallelepiped of that volume.
3. Ignore order up to signSwapping two vectors flips the sign but not the coplanarity verdict.

6Vector triple product identity

The vector triple product satisfies \vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}. The result lies in the plane of the inner two vectors — a useful expansion when a question nests a cross inside a cross.

Figure. The expansion \vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c} is a combination of \vec{b} and \vec{c}, so the result lies in their plane. Arrows are schematic, not a solid in space.

7Vector line and section formula

The vector equation of a line is \vec{r}=\vec{a}+\lambda\vec{b}. The position vector dividing the segment from \vec{a} to \vec{b} in the ratio m:n is \dfrac{n\vec{a}+m\vec{b}}{m+n} — the same weighted average as in coordinate geometry, written with vectors.

Figure. Segment from A to B with mid-point M (ratio 1:1). Schematic — not the numeric vectors of the ledger drawn to scale.