Lines in Space: Direction Cosines, Equations and Angles

Name: Lines in Space: Direction Cosines, Equations and Angles
Author: StemPy Lab

How do you tell a friend which way a kite string runs? You give a direction, and in space a direction needs three numbers. Learn direction cosines and direction ratios, write the equation of a line in vector and Cartesian form, and find the angle between two lines. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level Further Maths and IB.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Direction cosines and direction ratios
A direction in space needs three numbers. The direction cosines are the cosines of the angles that a line makes with the three axes. Direction ratios are any three numbers in the same proportion. Divide them by their root of the sum of squares and you get the direction cosines.
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
Equations of a line in space
A line in space is fixed by one point on it and its direction. In vector form it is $\vec{r} = \vec{a} + \lambda\vec{b}$ . In Cartesian form it is $\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}$ . The same line can be written in both ways, and also through two points.
1 simulation1 practice question1 Python program4 problems
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Lesson 3about 25 min
The angle between two lines
The angle between two lines is the angle between their directions. In terms of direction ratios, $\cos\theta = \dfrac{\lvert a_1a_2 + b_1b_2 + c_1c_2 \rvert}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}$ . The lines are perpendicular when the numerator is $0$ and parallel when the ratios are proportional.
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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Lines in Space: Practice Paper

15 questions · 20 min

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Lines in Space: Direction Cosines, Equations and Angles

Course outline (3 lessons)

Direction cosines and direction ratios

Equations of a line in space

The angle between two lines

Test Your Learning

Your teacher

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