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Vectors: Introductory Problems, Examples and MCQ Quizzes - Related to products, properties of vectors, proving geometric properties using vectors.






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                                                                                              Vectors: Introductory Problems

Here are some of the problems which have been solved in this tutorial (Also check out the MCQ Quizzes at the end of this page):

These problems cover the basic of vectors, products, properties of vectors and will also introduce you to the idea of proving geometric properties using vectors.


Q: Show that the sum of the squares of the sides of any quadrilateral (plane or skew) equals the sum of the squares of its diagonals together with four times the square of the line joining their middle points.

Q: In a triangle ABC, find the condition for AB2 + AC2 = 2 (AD2 + BD2) to be true where D is a point on BC.

Q: Prove that if a point is equidistant from the vertices of a right angled triangle, its joint to the middle point of the hypotenuse is perpendicular to the plane of the triangle.

Q: If a, b, c are mutually perpendicular vectors of equal magnitude, show that a + b + c is equally inclined to a, b, c.

Q: Given that a.b = a.c, a x b = a x c and a is a non-zero vector. Show that b=c.

Q: Prove that axb = [(i x a).b]i + [(j x a).b]j + [(k x a).b]k

Q: Show that the perpendicular distance of a point whose position vector is a from the plane through three points with position vectors b, c, d is [b c d] + [c a d] + [a b d] – [a b c]/[b x c + c x d + d x b]


Tutorial with Solved Problems (Also check out the two MCQ Quizzes at the end of this page) :



MCQ Quiz #1: Vectors

Companion MCQ Quiz #1 for Vectors- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

Vectors #1


MCQ Quiz #2: Vectors- More Challenging Problems

Companion MCQ Quiz #2 for Vectors- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

Quiz #2: The Basics of Vectors- More Challenging Problems



In case you're interested in learning more about Vectors, here's the full set of tutorials we have :

Vectors 1a ( Theory and Definitions: Introduction to Vectors; Vector, Scalar and Triple Products)
Introducing a vector, position vectors, direction cosines, different types of vectors, addition and subtraction of vectors. Vector and Scalar products. Scalar Triple product and Vector triple product and their properties. Components and projections of vectors.

Vectors 1b ( Solved Problem Sets: Introduction to Vectors; Vector, Scalar and Triple Products )
Solved examples and problem sets based on the above concepts. 



Vectors 2a ( Theory and Definitions: Vectors and Geometry ) Vectors and geometry. Parametric vectorial equations of lines and planes. Angles between lines and planes. Co-planar and collinear points. Cartesian equations for lines and planes in 3D. 

Vectors 2b ( Solved Problem Sets: Vectors and Geometry )

Solved examples and problem sets based on the above concepts.






Vectors 3a ( Theory and Definitions: Vector Differential and Integral Calculus ) Vector Differential Calculus. Derivative, curves, tangential vectors, vector functions, gradient, directional derivative, divergence and curl of a vector function; important formulas related to div, curl and grad. Vector Integral Calculus. Line integral, independence of path, Green's theorem, divergence theorem of Gauss, green's formulas, Stoke's theorems.


Vectors 3b ( Solved Problem Sets: Vector Differential and Integral Calculus ) - Solved examples and problem sets based on the above concepts.



 

 


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