Vectors and Their Algebra: Quick Revision

A brief note on Vectors and Their Algebra. Read it before a test. Try each question at the end before you open its answer.

Components and magnitude

• Position vector: $\overrightarrow{OP} = x\hat i + y\hat j + z\hat k$ for the point $P(x, y, z)$.
• Magnitude: $\lvert\vec a\rvert = \sqrt{a_1^2 + a_2^2 + a_3^2}$.
• Unit vector: $\hat a = \dfrac{\vec a}{\lvert\vec a\rvert}$.
• Direction cosines: $l = \cos\alpha = \dfrac{a_1}{\lvert\vec a\rvert}$, and $m$, $n$ in the same way. They satisfy $l^2 + m^2 + n^2 = 1$.

Algebra

• Sum and difference: add or subtract the matching components. The triangle law and the parallelogram law give the same result.
• Scalar multiple: $\lambda\vec a$ scales every component. Collinear vectors satisfy $\vec a = \lambda\vec b$.
• Triangle inequality: $\lvert\vec a + \vec b\rvert \le \lvert\vec a\rvert + \lvert\vec b\rvert$.
• Between two points: $\overrightarrow{AB} = \vec b - \vec a$, head minus tail.

The section formula

• Internal, $AP : PB = m : n$: $\vec p = \dfrac{n\vec a + m\vec b}{m + n}$.
• External: $\vec p = \dfrac{m\vec b - n\vec a}{m - n}$.
• Midpoint: $\dfrac{\vec a + \vec b}{2}$. Centroid: $\dfrac{\vec a + \vec b + \vec c}{3}$.
• Parallelogram $ABCD$: $\vec d = \vec a + \vec c - \vec b$.

A worked example

The point $P$ divides $A(1, 4, -2)$ and $B(6, -1, 8)$ internally in the ratio $3 : 2$. Find $P$.

1. The weights. Here $m = 3$ and $n = 2$. The point is closer to $A$ than to $B$ only if $m$ is the smaller number, so here it is closer to $B$.
2. The formula. $\vec p = \dfrac{2\vec a + 3\vec b}{5}$.
3. The coordinates. $\left(\dfrac{2 + 18}{5}, \dfrac{8 - 3}{5}, \dfrac{-4 + 24}{5}\right) = (4, 1, 4)$.

Practice with new numbers

A program to try

The program checks the worked example. It finds $P$, the ratio $AP : PB$, and tests that the three points lie on one line.

Mistakes to avoid

• Forgetting the square root in a magnitude.
• Adding magnitudes when the vectors do not point the same way.
• Swapping $m$ and $n$ in the section formula.
• Using tail minus head for the vector between two points.

Check yourself

1. Find the unit vector along $2\hat i - 6\hat j + 3\hat k$.

2. Find $\overrightarrow{AB}$ for $A(3, 1, -2)$ and $B(0, 5, 4)$.

3. Find the midpoint of the points in question 2.

4. Find the point dividing $A(0, 0, 0)$ and $B(9, 6, 3)$ internally in the ratio $1 : 2$.

5. Two vectors have magnitudes $5$ and $12$. What are the smallest and largest possible magnitudes of their sum?

Sources

• NCERT, Mathematics Part II for Class XII, Chapter 10, sections on basic concepts, addition of vectors and the section formula.
• OpenStax, Calculus Volume 3, Chapter 2, sections on vectors in the plane and in three dimensions.