Applications of Vector Algebra: One Mark Questions, Book Back Answers and Notes

About Applications of Vector Algebra

Hard ~180 min study

A vector carries two pieces of information, a magnitude and a direction, and that is exactly what you need to describe a line or a plane in space. In Class 11 you learnt the dot product and the cross product. This unit puts those two products to work.

The chapter starts with triple products. The scalar triple product gives the volume of a parallelepiped and tells you whether three vectors are coplanar. The vector triple product has its own expansion formula, which you must remember exactly. After that, the same ideas are used to write the equation of a straight line through a point in a given direction, and the equation of a plane with a given normal. Once you have these equations, finding the angle between two lines, the distance between two skew lines or the foot of a perpendicular becomes a matter of applying a formula carefully.

When you revise, practise moving between forms. A question may give a line in Cartesian form and expect the answer in vector form, or give a plane in normal form and ask for its intercepts. Vector proofs of the cosine and sine formulas, and problems on work done and torque, also come from this unit.

What you'll learn

• Use the scalar triple product to find volumes and test whether vectors are coplanar.
• Prove the cosine and sine formulas and other trigonometric results with vectors.
• Find the work done by a force and the torque of a force about a point.
• Write the equation of a line in parametric, non-parametric and Cartesian form and convert between them.
• Write the equation of a plane in normal, intercept and point-normal form.
• Calculate angles and distances between lines and planes in space.

Before you start

• Addition of vectors and multiplication of a vector by a scalar.
• The dot product and cross product and their basic properties.
• Coordinate geometry in two dimensions and basic trigonometry.

Applications of Vector Algebra explained

What This Unit Covers

Triple Products and What They Measure

The scalar triple product $[ \vec{a}, \vec{b}, \vec{c} ] = (\vec{a} \times \vec{b}) \cdot \vec{c}$ is a number. Its absolute value is the volume of the parallelepiped with $\vec{a}$, $\vec{b}$ and $\vec{c}$ as coterminous edges. If the product is zero, the volume is zero, so the three vectors are coplanar. In practice you compute it as a 3 by 3 determinant of the components.

The vector triple product $\vec{a} \times (\vec{b} \times \vec{c})$ is a vector, and it expands as $(\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$. The result lies in the plane of $\vec{b}$ and $\vec{c}$. The cross product is not associative, so $(\vec{a} \times \vec{b}) \times \vec{c}$ is generally a different vector. Students often apply the same expansion to both groupings and get the wrong answer. The textbook also uses Lagrange's identity and the Jacobi identity, both of which follow from this expansion.

Trigonometry and Mechanics

Take the sides of a triangle as vectors and square them using the dot product: the cosine formula falls out in two lines. Using the cross product, whose magnitude for two sides is twice the area of the triangle, you get the sine formula. Compound angle formulas such as $\cos(\alpha - \beta)$ are proved the same way, using unit vectors at angles $\alpha$ and $\beta$ to the x-axis.

In mechanics, the work done by a force $\vec{F}$ that moves a particle through a displacement $\vec{d}$ is $W = \vec{F} \cdot \vec{d}$, a scalar. The torque of a force about a point is $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector of the point of application relative to that point. Torque is a vector, and the order $\vec{r} \times \vec{F}$ matters.

Straight Lines in Space

A line is fixed by one point on it and one direction. If $\vec{a}$ is the position vector of the point and $\vec{b}$ is parallel to the line, the parametric vector equation is $\vec{r} = \vec{a} + t\vec{b}$. Writing $\vec{b}$ in components gives the Cartesian form, where the denominators are the direction ratios. Removing the parameter gives the non-parametric form $(\vec{r} - \vec{a}) \times \vec{b} = \vec{0}$. The line through two points is handled the same way, with $\vec{b}$ taken as the difference of their position vectors.

The angle between two lines is the angle between their direction vectors. Two lines that are not parallel and do not meet are called skew lines. For them the shortest distance is the length of the common perpendicular, found using the scalar triple product. If that distance is zero, the lines intersect.

Planes

A plane is fixed by a normal vector and either a point on the plane or its distance from the origin. The normal form is $\vec{r} \cdot \hat{d} = p$, where $\hat{d}$ is the unit normal and $p$ is the perpendicular distance from the origin. The intercept form $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ shows directly that the plane cuts the axes at $a$, $b$ and $c$. The chapter also covers the plane through three points, the plane containing two lines, and the condition for two lines to be coplanar.

The last part applies these equations: the angle between two planes, the angle between a line and a plane, the point where a line meets a plane, the distance of a point from a plane, the distance between two parallel planes, the foot of the perpendicular and the image of a point in a plane.

Common mistakes to avoid

• Mixing up the two triple products. The scalar triple product gives a number, the vector triple product gives a vector.
• Reading the direction ratios from a Cartesian equation that is not in standard form, for example when x has a coefficient like 2x in the numerator.
• Taking the constant q in r · n = q as the distance from the origin without first dividing by the magnitude of n.
• Using cosine for the angle between a line and a plane. The angle between the line and the normal is its complement, so the formula uses sine.
• Applying the skew-lines distance formula to parallel lines, where the cross product of the direction vectors is zero and a different formula is needed.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is the scalar triple product used for?

Its absolute value gives the volume of the parallelepiped formed by three vectors as adjacent edges. If the product is zero, the three vectors are coplanar. The same test tells you whether four points lie in one plane.

How are skew lines different from parallel lines?

Both never meet. Parallel lines have the same direction and lie in one plane. Skew lines have different directions and do not lie in any common plane, which can only happen in three dimensions.

How do I find the angle between a line and a plane?

Find the angle between the direction vector of the line and the normal of the plane. The required angle is its complement, so sin θ equals the absolute value of the dot product of the two vectors divided by the product of their magnitudes.

What does the intercept form of a plane tell me?

In x/a + y/b + z/c = 1, the plane meets the x, y and z axes at a, b and c. To use it, first rewrite the equation so that the right side is 1.

Why does a line have more than one form of equation?

Each form suits a different job. The parametric form gives any point on the line by choosing a value of t, which helps in finding intersections and feet of perpendiculars. The Cartesian form shows the direction ratios at a glance.

What does the shortest distance between two lines tell me?

If the shortest distance is zero, the lines intersect. If it is not zero and the lines are not parallel, they are skew, and the value tells you how close they come.