6.0 REVIEW : INTRODUCTION TO VECTORS
Definition:
A vector is a quantity which has both magnitude and direction.
Examples: displacement, velocity, acceleration, force.
A scalar has only magnitude.
Examples: mass, time, distance, temperature.
Direction in Real Life:
Aeroplane flying: velocity has direction and speed.
Aeroplane landing: approach direction aligns with runway.
Wind flow, river current, GPS navigation use vectors.
Position Vector:
Position vector of point A(x, y) from origin O is:
OA = xi + yj
Unit Vectors:
i is unit vector along x-axis
j is unit vector along y-axis
Magnitude of i = 1
Magnitude of j = 1
Any vector a = xi + yj
Types of Vectors:
Zero Vector:
Magnitude is zero.
Coordinates (0, 0).
No fixed direction.
Unit Vector:
Magnitude is one.
Examples: i, j
Equal Vectors:
Same magnitude and same direction.
Collinear Vectors:
Parallel or anti-parallel vectors.
Negative Vector:
Same magnitude but opposite direction.
Position Vector:
Vector drawn from origin to a point.
Applications of Vectors:
Navigation and GPS
Physics (forces, motion)
Engineering structures
Games and computer graphics
Products of Vectors:
Scalar (Dot) product gives a scalar.
Vector (Cross) product gives a vector (not included here).
Projection is the component of one vector on another.
6.1 SCALAR (DOT) PRODUCT
Definition:
For vectors a and b with angle theta between them,
a dot b = |a| |b| cos(theta)
Result is a scalar.
Geometrical Meaning:
a dot b = magnitude of a multiplied by projection of b on a.
If theta = 90 degrees, cos 90 = 0
Therefore a dot b = 0
If a dot b = 0, vectors are perpendicular (unless one vector is zero).
Dot Product of Unit Vectors:
i dot i = 1
j dot j = 1
i dot j = 0
j dot i = 0
Component Form:
If a = (x1, y1) and b = (x2, y2)
a dot b = x1x2 + y1y2
Magnitude of Vector:
|a| squared = a dot a = x1 squared + y1 squared
|a| = square root of (x1 squared + y1 squared)
Angle Between Two Vectors:
cos(theta) = (a dot b) / (|a| |b|)
Maximum when theta = 0 degrees
Minimum when theta = 180 degrees
Properties of Dot Product:
a dot b = b dot a
a dot (b + c) = a dot b + a dot c
(a + b) squared = a squared + 2a dot b + b squared
(a - b) squared = a squared - 2a dot b + b squared
(a + b)(a - b) = a squared - b squared
6.2 VECTOR GEOMETRY
Midpoint Formula:
Midpoint of A(a) and B(b) is (a + b) / 2
Section Formula:
Internal Division (ratio m : n):
Position vector of P = (m b + n a) / (m + n)
External Division (ratio m : n):
Position vector of P = (m b - n a) / (m - n)
Centroid of Triangle:
If position vectors of vertices are a, b, c
Centroid G = (a + b + c) / 3
6.3 THEOREMS ON TRIANGLES
Midpoint Theorem:
Line joining midpoints of two sides of triangle is parallel to third side and half its length.
Isosceles Triangle:
Median to base is perpendicular to base.
Right Triangle:
Midpoint of hypotenuse is equidistant from all vertices.
6.4 QUADRILATERALS AND SEMICIRCLE
Varignon’s Theorem:
Joining midpoints of quadrilateral forms a parallelogram.
Parallelogram:
Diagonals bisect each other.
Rhombus:
Diagonals bisect at right angles.
Rectangle:
Diagonals are equal.
Semicircle Theorem:
Angle in semicircle is 90 degrees.
IMPORTANT PRACTICE QUESTIONS WITH SOLUTIONS
Question 1:
Define scalar product.
Solution:
Scalar product of vectors a and b is given by:
a dot b = |a| |b| cos(theta)
Question 2:
If a dot b = 0, what can you conclude?
Solution:
Vectors a and b are perpendicular.
Question 3:
Find dot product of a = (2, 3) and b = (4, -1)
Solution:
a dot b = 2(4) + 3(-1)
= 8 - 3
= 5
Question 4:
Find magnitude of vector a = (5, 12)
Solution:
|a| = square root of (5 squared + 12 squared)
= square root of (25 + 144)
= square root of 169
= 13
Question 5:
Find angle between a = (1, 2) and b = (2, 1)
Solution:
a dot b = 1(2) + 2(1) = 4
|a| = square root of 5
|b| = square root of 5
cos(theta) = 4 / 5
theta = cos inverse (4/5)
Question 6:
Find midpoint of A(1, 3) and B(5, 7)
Solution:
Midpoint = ((1 + 5)/2, (3 + 7)/2)
= (3, 5)
Question 7:
Find position vector of point dividing A(2, 4) and B(8, 10) internally in ratio 1 : 2
Solution:
Let a = (2, 4), b = (8, 10)
P = (1b + 2a) / 3
= ((8,10) + (4,8)) / 3
= (12,18) / 3
= (4,6)
Question 8:
If a = (3, 4) and b = (6, 8), check whether vectors are parallel.
Solution:
b = 2a
Therefore vectors are parallel.
Question 9:
Prove that diagonals of parallelogram bisect each other.
Solution:
Using position vectors of vertices, midpoint of both diagonals is same.
Question 10:
In a right triangle, prove midpoint of hypotenuse is equidistant from all vertices.
Solution:
Using dot product, distances from midpoint to all vertices are equal.
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