Applications of Vector Algebra - Study Notes

Applications of Vector Algebra — Study Notes

This chapter pushes the dot and cross products one step further: it combines three vectors at a time, then uses those combinations to describe lines and planes in three‑dimensional space. The notes below summarise every idea you need for the book‑back exercise.

1. Scalar Triple Product (the box product)

For vectors \(\vec{a},\vec{b},\vec{c}\), the scalar \((\vec{a}\times\vec{b})\cdot\vec{c}\) is called the scalar triple product and is written \([\vec{a},\vec{b},\vec{c}]\). In components it is a determinant:

\[[\vec{a},\vec{b},\vec{c}]=\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}.\]

• Volume: \(\big|[\vec{a},\vec{b},\vec{c}]\big|\) equals the volume of the parallelepiped built on the three vectors as coterminous edges.
• Cyclic symmetry: \([\vec{a},\vec{b},\vec{c}]=[\vec{b},\vec{c},\vec{a}]=[\vec{c},\vec{a},\vec{b}]\). Interchanging any two vectors multiplies the value by \(-1\).
• Dot and cross swap: \((\vec{a}\times\vec{b})\cdot\vec{c}=\vec{a}\cdot(\vec{b}\times\vec{c})\) — the dot and cross can trade places without changing the value.
• Coplanarity test: \([\vec{a},\vec{b},\vec{c}]=0\) precisely when the three vectors lie in one plane (any one is a linear combination of the other two).
• Repeated vector: if any two of the three vectors are equal or parallel, the box product is \(0\).

2. Vector Triple Product

The product \(\vec{a}\times(\vec{b}\times\vec{c})\) is a vector, expanded by the BAC–CAB rule:

\[\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\,\vec{b}-(\vec{a}\cdot\vec{b})\,\vec{c}.\]

The result always lies in the plane of the two bracketed vectors \(\vec{b}\) and \(\vec{c}\). The operation is not associative: in general \(\vec{a}\times(\vec{b}\times\vec{c})\neq(\vec{a}\times\vec{b})\times\vec{c}\). A handy mirror form is \((\vec{a}\times\vec{b})\times\vec{c}=(\vec{a}\cdot\vec{c})\,\vec{b}-(\vec{b}\cdot\vec{c})\,\vec{a}\).

3. Jacobi’s and Lagrange’s Identities

• Jacobi: \(\vec{a}\times(\vec{b}\times\vec{c})+\vec{b}\times(\vec{c}\times\vec{a})+\vec{c}\times(\vec{a}\times\vec{b})=\vec{0}\).
• Lagrange: \((\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})=(\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d})-(\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c})\).
• Useful corollary: \([\vec{a}\times\vec{b},\ \vec{b}\times\vec{c},\ \vec{c}\times\vec{a}]=[\vec{a},\vec{b},\vec{c}]^{2}\). This single identity unlocks several MCQs about volumes of parallelepipeds built from cross products.

4. Equations of a Straight Line

A line is fixed either by a point and a direction, or by two points.

• Point \(\vec{a}\) and direction \(\vec{b}\): parametric vector form \(\vec{r}=\vec{a}+t\vec{b}\); non‑parametric form \((\vec{r}-\vec{a})\times\vec{b}=\vec{0}\).
• Cartesian form: \(\dfrac{x-x_{1}}{b_{1}}=\dfrac{y-y_{1}}{b_{2}}=\dfrac{z-z_{1}}{b_{3}}\), where \((b_{1},b_{2},b_{3})\) are direction ratios.
• Two points \(\vec{a},\vec{b}\): \(\vec{r}=\vec{a}+t(\vec{b}-\vec{a})\); Cartesian form uses direction ratios \((x_{2}-x_{1},\,y_{2}-y_{1},\,z_{2}-z_{1})\).
• Direction cosines \(l,m,n\) satisfy \(l^{2}+m^{2}+n^{2}=1\) and are proportional to the direction ratios.

5. Relations Between Two Lines

• Angle: for directions \(\vec{b}\) and \(\vec{d}\), \(\cos\theta=\dfrac{|\vec{b}\cdot\vec{d}|}{|\vec{b}|\,|\vec{d}|}\).
• Parallel when \(\vec{b}=\lambda\vec{d}\); perpendicular when \(\vec{b}\cdot\vec{d}=0\).
• Coplanar vs. skew: lines that are parallel or intersecting are coplanar; otherwise they are skew (non‑coplanar).
• Shortest distance (skew lines) \(\vec{r}=\vec{a}+s\vec{b}\) and \(\vec{r}=\vec{c}+t\vec{d}\): \(d=\dfrac{\big|(\vec{c}-\vec{a})\cdot(\vec{b}\times\vec{d})\big|}{|\vec{b}\times\vec{d}|}\). The lines intersect (are coplanar) when \((\vec{c}-\vec{a})\cdot(\vec{b}\times\vec{d})=0\).
• Shortest distance (parallel lines) \(\vec{r}=\vec{a}+s\vec{b}\) and \(\vec{r}=\vec{c}+t\vec{b}\): \(d=\dfrac{\big|(\vec{c}-\vec{a})\times\vec{b}\big|}{|\vec{b}|}\).
• Foot of perpendicular: take a general point on the line, form the vector to the external point, and set its dot product with the direction to zero.

6. Equations of a Plane

• Normal form: \(\vec{r}\cdot\hat{d}=p\) (unit normal \(\hat{d}\), distance \(p\)); Cartesian \(lx+my+nz=p\).
• Point \(\vec{a}\) and normal \(\vec{n}\): \((\vec{r}-\vec{a})\cdot\vec{n}=0\), i.e. \(a(x-x_{1})+b(y-y_{1})+c(z-z_{1})=0\).
• Intercept form: \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\).
• Three non‑collinear points: \([\vec{r}-\vec{a},\ \vec{b}-\vec{a},\ \vec{c}-\vec{a}]=0\).
• Point and two parallel vectors: \(\vec{r}=\vec{a}+s\vec{b}+t\vec{c}\); non‑parametric \([\vec{r}-\vec{a},\ \vec{b},\ \vec{c}]=0\).
• General plane \(ax+by+cz+d=0\) has normal \((a,b,c)\).

7. Angles, Containment and Intersections

• Angle between planes with normals \(\vec{n}_{1},\vec{n}_{2}\): \(\cos\theta=\dfrac{|\vec{n}_{1}\cdot\vec{n}_{2}|}{|\vec{n}_{1}|\,|\vec{n}_{2}|}\).
• Angle between a line and a plane (direction \(\vec{b}\), normal \(\vec{n}\)): \(\sin\theta=\dfrac{|\vec{b}\cdot\vec{n}|}{|\vec{b}|\,|\vec{n}|}\). Note the sine, not cosine, because the plane’s angle is the complement of the angle with the normal.
• Line lies in a plane when the point on the line satisfies the plane and the direction is perpendicular to the normal (\(\vec{b}\cdot\vec{n}=0\)).
• Two lines are coplanar when \((\vec{c}-\vec{a})\cdot(\vec{b}\times\vec{d})=0\).
• Meeting point of a line and a plane: substitute the parametric point of the line into the plane equation, solve for the parameter, and back‑substitute.
• Plane through the intersection of two planes: \((\vec{r}\cdot\vec{n}_{1}-d_{1})+\lambda(\vec{r}\cdot\vec{n}_{2}-d_{2})=0\); fix \(\lambda\) using an extra condition (a point or perpendicularity).

8. Distances and Images

• Point to plane: \(\delta=\dfrac{|\vec{u}\cdot\vec{n}-p|}{|\vec{n}|}\) or \(\delta=\dfrac{|ax_{1}+by_{1}+cz_{1}-p|}{\sqrt{a^{2}+b^{2}+c^{2}}}\).
• Origin to plane \(ax+by+cz+d=0\): \(\delta=\dfrac{|d|}{\sqrt{a^{2}+b^{2}+c^{2}}}\).
• Two parallel planes \(ax+by+cz+d_{1}=0\) and \(ax+by+cz+d_{2}=0\): \(\delta=\dfrac{|d_{1}-d_{2}|}{\sqrt{a^{2}+b^{2}+c^{2}}}\). First make the normals identical.
• Image of a point \(\vec{u}\) in the plane \(\vec{r}\cdot\vec{n}=p\): \(\vec{v}=\vec{u}+\dfrac{2\,[\,p-(\vec{u}\cdot\vec{n})\,]}{|\vec{n}|^{2}}\,\vec{n}\); the midpoint of \(\vec{u}\) and \(\vec{v}\) is the foot of the perpendicular.

9. Common Exam Traps

• The box product can be negative; a volume is its absolute value. Watch for questions that ask for the signed value versus the magnitude.
• For a line and a plane use \(\sin\theta\); for two planes or two lines use \(\cos\theta\). Mixing them up is the most frequent slip.
• Before using the parallel‑planes distance formula, rescale so both equations have identical normal coefficients.
• Re‑cast a Cartesian line such as \(\dfrac{2y+3}{3}\) into the standard \(\dfrac{y-y_{1}}{b_{2}}\) form before reading off direction ratios.
• The vector triple product is not associative — keep the brackets where they are.
• When checking that a line lies in a plane, both conditions must hold; perpendicularity of direction and normal alone is not enough.