Magnitude and unit vector
Vector Algebra · Vectors and Their Types\[|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2},\qquad \hat a=\frac{\vec a}{|\vec a|}\]
Chapter 10 · Vectors and Three-Dimensional Geometry
Vector Algebra
Chapter Overview
Many quantities – displacement, velocity, force – have both a magnitude and a direction. Vectors represent them. In this chapter you will learn the types of vectors, how to add them and multiply them by scalars, how to work with components \(a\hat i+b\hat j+c\hat k\), the section formula, and the two products: the scalar (dot) product, which measures angles and projections, and the vector (cross) product, which gives perpendicular directions and areas.
Connection: Vectors are the language of Three-Dimensional Geometry (next chapter).
Topics
Key Concepts
- \(\vec a=a_1\hat i+a_2\hat j+a_3\hat k\), \(|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2}\), unit vector \(\hat a=\frac{\vec a}{|\vec a|}\).
- Section formula: \(\vec r=\frac{m\vec b+n\vec a}{m+n}\) (internal, ratio \(m:n\)).
- \(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta=a_1b_1+a_2b_2+a_3b_3\); \(\vec a\perp\vec b\iff\vec a\cdot\vec b=0\).
- \(|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta\) = area of the parallelogram; \(\vec a\parallel\vec b\iff\vec a\times\vec b=\vec0\).
Formulas
Section formula
Vector Algebra · Components and Section Formula\[\vec r=\frac{m\vec b+n\vec a}{m+n}\]
Dot product and projection
Vector Algebra · Scalar (Dot) Product\[\vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3,\quad \text{proj}_{\vec b}\vec a=\frac{\vec a\cdot\vec b}{|\vec b|}\]
Cross product and areas
Vector Algebra · Vector (Cross) Product\[|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta,\quad \Delta=\tfrac12|\vec a\times\vec b|\]
Key Points
Direction cosines
Vector Algebra · Vectors and Their TypesFor \(\vec r=x\hat i+y\hat j+z\hat k\): \(l=\frac xr,\ m=\frac yr,\ n=\frac zr\) and \(l^2+m^2+n^2=1\).
Useful identities
Vector Algebra · Scalar (Dot) Product\(|\vec a+\vec b|^2=|\vec a|^2+|\vec b|^2+2\vec a\cdot\vec b\); \((\vec a+\vec b)\cdot(\vec a-\vec b)=|\vec a|^2-|\vec b|^2\).
Common Mistakes
Section formula weights
Vector Algebra · Components and Section FormulaFor ratio \(m:n\) from \(A\) to \(B\), the weight \(m\) goes with \(\vec b\) (the far point).
Projection formula
Vector Algebra · Scalar (Dot) ProductProjection of \(\vec a\) on \(\vec b\) divides by \(|\vec b|\), not \(|\vec a|\).
Order of the cross product
Vector Algebra · Vector (Cross) Product\(\hat j\times\hat i=-\hat k\), not \(\hat k\). Reversing the order changes the sign.
Solved Examples
Area of a triangle
Vector Algebra · Vector (Cross) ProductVertices \(A(0,1,1),B(1,2,3),C(2,0,1)\).
\(\overrightarrow{AB}=\hat i+\hat j+2\hat k\), \(\overrightarrow{AC}=2\hat i-\hat j\). \(\overrightarrow{AB}\times\overrightarrow{AC}=2\hat i+4\hat j-3\hat k\), area \(=\frac12\sqrt{29}\).
Practice & Tests
Chapter Practice
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