Chapter 10

Vector Algebra

Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.

Interactive lesson All chapters Past paper questions (48)

Examples

Example 1Displacement of 40 km, 30° west of south, drawn to scaleworked
Example 2Scalar or vector: 5 seconds, 1000 cm³, 10 Newton, 30 km/hr, 10 g/cm³, 20 m/s northworked
Example 3Collinear, equal and coinitial vectors in Fig 10.5worked
Example 4 $x, y, z$ making $x\hat i+2\hat j+z\hat k$ equal to $2\hat i+y\hat j+\hat k$ worked
Example 5 $\hat i+2\hat j$ and $2\hat i+\hat j$ ⁠: equal magnitudes, unequal vectorsworked
Example 6Unit vector along $2\hat i+3\hat j+\hat k$ worked
Example 7Vector of magnitude 7 along $\hat i-2\hat j$ worked
Example 8Unit vector along the sum of $2\hat i+2\hat j-5\hat k$ and $2\hat i+\hat j+3\hat k$ worked
Example 9Direction ratios and direction cosines of $\hat i+\hat j-2\hat k$ worked
Example 10 $\overrightarrow{PQ}$ for $P(2,3,0)$ ⁠, $Q(-1,-2,-4)$ worked
Example 11Point dividing $3\vec a-2\vec b$ ⁠, $\vec a+\vec b$ in the ratio 2 : 1, internally and externallyworked
Example 12Right angled triangle with vertices $2\hat i-\hat j+\hat k$ ⁠, $\hat i-3\hat j-5\hat k$ ⁠, $3\hat i-4\hat j-4\hat k$ worked
Example 13Angle between $\vec a$ ⁠, $\vec b$ with $|\vec a|=1$ ⁠, $|\vec b|=2$ ⁠, $\vec a\cdot\vec b=1$ worked
Example 14Angle between $\hat i+\hat j-\hat k$ and $\hat i-\hat j+\hat k$ worked
Example 15 $\vec a+\vec b\perp\vec a-\vec b$ for $5\hat i-\hat j-3\hat k$ ⁠, $\hat i+3\hat j-5\hat k$ worked
Example 16Projection of $2\hat i+3\hat j+2\hat k$ on $\hat i+2\hat j+\hat k$ worked
Example 17 $|\vec a-\vec b|$ when $|\vec a|=2$ ⁠, $|\vec b|=3$ ⁠, $\vec a\cdot\vec b=4$ worked
Example 18 $|\vec x|$ when $(\vec x-\vec a)\cdot(\vec x+\vec a)=8$ ⁠, $\vec a$ a unit vectorworked
Example 19Cauchy-Schwarz inequality $|\vec a\cdot\vec b|\le|\vec a||\vec b|$ worked
Example 20Triangle inequality $|\vec a+\vec b|\le|\vec a|+|\vec b|$ worked
Example 21Collinear points $A(-2\hat i+3\hat j+5\hat k)$ ⁠, $B(\hat i+2\hat j+3\hat k)$ ⁠, $C(7\hat i-\hat k)$ worked
Example 22 $|\vec a\times\vec b|$ for $2\hat i+\hat j+3\hat k$ ⁠, $3\hat i+5\hat j-2\hat k$ worked
Example 23Unit vector perpendicular to $\vec a+\vec b$ and $\vec a-\vec b$ ⁠, $\vec a=\hat i+\hat j+\hat k$ ⁠, $\vec b=\hat i+2\hat j+3\hat k$ worked
Example 24Area of the triangle $A(1,1,1)$ ⁠, $B(1,2,3)$ ⁠, $C(2,3,1)$ worked
Example 25Area of the parallelogram on $3\hat i+\hat j+4\hat k$ and $\hat i-\hat j+\hat k$ worked
Example 26All the unit vectors in the XY-plane: $\cos\theta\,\hat i+\sin\theta\,\hat j$ worked
Example 27Angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is $\pi$ ⁠; they are collinearworked
Example 28 $|\vec a+\vec b+\vec c|$ when each is perpendicular to the sum of the other twoworked
Example 29 $\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a$ when $\vec a+\vec b+\vec c=\vec 0$ ⁠, $|\vec a|=3$ ⁠, $|\vec b|=4$ ⁠, $|\vec c|=2$ worked
Example 30Split $\vec\beta=2\hat i+\hat j-3\hat k$ into parts parallel and perpendicular to $\vec\alpha=3\hat i-\hat j$ worked

Exercise 10.1

Exercise 10.2

Exercise 10.3

Exercise 10.4

Miscellaneous Exercise

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