Vector properties: Vector geometry is a fundamental idea in mathematics that connects algebra and geometry. It offers an effective foundation for comprehending shapes, transformations, and spatial relationships.

What are vectors?

A vector is a mathematical entity that has both magnitude and direction.Vectors offer more details regarding movement and location than scalars, which are only defined by magnitude. Vectors are often represented by the following symbols and can be shown in two or three dimensions:

• 2D vector : v = (x,y)
• 3D vector : v= (x,y,z)

Vectors are often drawn as arrows , where the length represents magnitude and the arrowhead indicates the direction.Vector properties are very important.

Basic operations on vectors

1 . Vector Addition

Vector addition follows the paralelogram law or the triangle rule. If two vectors a=(x1​,y1) and b=(x2​,y2​) are given, their sum is

​In graphical representation, placing the tail of b at head of a results in the sum vector a+b, which extends from the tail of a to the head of b.

2. Scalar multiplication

Multiplying a vector v=(x,y) by a scalar k results in

This scales the vector while maintaining its direction.

3. Dot product (scalar product)

The dot product of two vectors a=(x1​,y1) and b=(x2​,y2​) is given by

It is used to find angles between vectors and check for perpendicularity.

4. Cross product (vector product): For 3D vectors

For two vectors, a=(x1​,y1​,z1​), b= (x2​,y2​,z2​), the cross products result in a new vector 

Where i,j,k are the unit vectors along the x,y and z axes, respectively

Expand the determinant

using cofactor expansion along the first row

Now, compute the 2×2 determinants

Construct the cross-product  Vector

Important  properties of  Vector

1. Orthogonality property: a⋅b=0 if and only if a⊥b

The dot product is zero when vectors are perpendicular

2. Parallel vector property: a×b=0 If and only if a∥b

The cross product is zero when vectors are parallel.

3. Dot product formula: a⋅b=∣a∣∣b∣cosθ

Where θ is the angle between two vectors .

4. Cross-product formula: a×b=∣a∣∣b∣sinθ n

Where θ is the angle between a and b and n is the unit vector perpendicular to both.

5. Unit Vector: 

A unit vector has a magnitude of 1 and represents direction.

6. Angle between two vectors: 

Used to find the angle between two vectors.

7. Projection of one vector onto another :

Find the component of a along b.

8. Commutative property: 

The order of the dot product does not matter.

9. Anti-commutative property:

10. Area of a parallelogram: 

                                                                                                                  Area=∣a×b∣

The magnitude of the cross product gives the area of the parallelogram spanned by a and b.

11. Volume of a parallelopiped: 

                                                                                                                 Volume=∣a⋅(b×c)∣

The scalar triple product gives the volume of the parallelopiped formed by three vectors.

Example 1:

Use the dot product to find the angles of the triangles whose vertices are the paints (1,3,2), (2,-1,1) and (-1,2,3)

Finding the angles of a triangle using the dot product.

Solution:

We are given the three vertices of a triangle in three-dimensional space:

A = (1,3,2), B = (2,-1,1), C= (-1,2,3)

Step 1 : Find the Vectors representing the sides

To find the angles, we first determine the vectors along the sides of the triangle.

Vector along the sides

Step 2 : Use the dot product formula to find angles 

Step 3: Compute each angle 

Angle at A (Between )

First compute the dot product

Now compute the magnitude

Angle at B (Between )

Magnitudes

Angle at C (Between)

Magnitudes

Final angles of the triangle

• θ_A​≈95.52∘
• θ_B​≈148.68∘
• θ_C​≈115.80∘

Example 2

If the three points are A(2,4,-1) B(0,2,-3) C(3,2,1), then find a vector perpendicular to the plane ABC

Solution:

Step 1 : Find two vectors in the plane 

WE can define two vectors in the plane using the given points

AB=B-A

AC=C-A

Computing AB:

AB= (0-2, 2-4,  -3-(-1))= (-2,-2,-2)

Computing AC:

AC= (3-2,2-4,1-(-1)) =(1,-2,2)

Step 2: Compute the Cross product AB×AC

Expanding the determinant

Step 2: Compute the determinants

For i

For j (note the negative side outside):

-(-2)=2

For k

Step 4: Final normal vector

 

Check out vector space

Two dimensional geometry formulas  sometimes known as plane geometry formulas , are the branch of geometry concerned with shapes, figures, and their properties in the two-dimensional plane. The plane has two dimensions: length and width, represented by the x-axis (horizontal) and y-axis (vertical) in a Cartesian coordinate system.

The basic ideas of two-dimensional geometry are points, lines, angles, forms (such as triangles, circles, and polygons), and their attributes. Let’s look at some essential aspects:

1. Cartesian Coordinates and Plane

The Cartesian coordinate system is the basis for two-dimensional geometry. The coordinate plane is a flat surface with points specified by a pair of numbers  (x,y).

•  x represents the abscissa (horizontal distance from the origin).
•  y represents the ordinate (vertical distance from the origin).

The origin (0,0) is where the x-axis and y-axis cross.

2. Points and coordinates

• A point is an location in the plane without size or dimension, expressed by coordinates  (x,y).
• The distance between two points (x_1, y_1​) and  (x_2, y_2) is given by the distance formula:

• The midpoint between two points (𝑥_1, 𝑦_ 1)  and (𝑥_ 2, 𝑦_ 2) is the point that divides the segment connecting them into two equal parts.

3. Lines and slopes

A line in the plane can be represented by the following linear equation:

• The slope (𝑚) of a line running through two points (𝑥_1, 𝑦_1)  and (𝑥_2, 𝑦_2)  can be calculated as:

• The equation of a line in slope-intercept form (when the slope $m$ and y-intercept c are known) is:

4. Angles

Angle between two lines: The  angle θ between two lines with slopes of m_1  and 𝑚_2  is as follows:

• The angle between two parallel lines is 0∘, and they both have the same slope.
• The slopes of perpendicular lines are negative reciprocals of one another, meaning that 𝑚_1 𝑚_2 = −1 , and the angle between them is 90∘ .

5. Distance between a point and a line.

The distance between a point P(x_0​,y_0​) and a line Ax+By+C=0 is determined by the formula:

6. Types of 2D Triangle Geometric Figures

A triangle is a three-sided polygon. Triangles can be classified according to angles as follows:

• Acute triangle: All angles less than 90∘
• Right triangle: One angle is 90∘
• Obtuse triangle: One angle is greater than 90∘

A triangle’s area is:

The area of a triangle with vertices A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), and  is as follows:

Quadrilaterals
A four-sided polygon is called a quadrilateral. Among the varieties are:

Square , Rectangular, Parallelogram, trapezium, rhombus
A rectangle’s area is:

Circles
The set of all points in the plane that are equally spaced from a fixed point (the center) is called a circle. The equation for a circle with radius r and center (h,k) is as follows:

The area and circumference of a circle are given by:

Area:

Circumference:

Polygons
A closed figure with straight sides is called a polygon. Among the examples are:

Pentagon (5 sides)
Hexagon (6 sides)
Octagon (8 sides)

The sum of the interior angles of a polygon with $n$ sides is given by:

Sum of interior angles=(n−2)×180∘

7. Conic Sections

Conic sections are curves that can be obtained by intersecting a plane with a cone. Each has a specific general equation.

• Circle:

• Ellipse:

• Parabola:

• Hyperbola:

8. Coordinated Geometry

In Two dimensional geometry formulas , coordinate geometry (also known as analytic geometry) employs the Cartesian coordinate system to algebraically represent and investigate geometric shapes.

For example:

• The distance formula finds the distance between two places.
• The slope of a line indicates its steepness and direction.
• A circle’s equation can be calculated using the distance between the center and a point on the circle.

Cartesian to polar equations: The Cartesian coordinate system and the Polar coordinate system are two different ways of representing points in a plane. In this blog, we’ll explain how to convert Cartesian coordinates (x, y) to Polar coordinates (r, θ) and vice versa, with examples to help understand.

1. Cartesian to Polar Conversion Examples

Quadrant I:

• Cartesian coordinates: (3,4)
  • Find r:

• Find θ:

Polar coordinates: (r=5,θ=0.93) or (r=5,θ=53.13∘)

Quadrant II:

• Cartesian coordinates: (−3,4)
  • Find r:

• Find θ:

Since the point is in the second quadrant, we need to add 𝜋 (or 180°) to achieve the correct angle.

Polar coordinates: (r=5,θ=2.21) or (r=5,θ=126.87∘)

Quadrant III:

• Cartesian coordinates(−3,−4)

• Find r

• Find θ:

Since the point is in the third quadrant, we need to add 𝜋 (or 180°) to the angle.

Polar coordinates: (r=5,θ=2.21) or (r=5,θ=126.87∘)

Quadrant IV:

• Cartesian coordinates: $(3,-4)$
  • Find r

• Find θ

As the point is in the fourth quadrant, the angle is already correct:

• Polar coordinates: (r=5,θ=−0.93) or (r=5,θ=−53.13∘)

The following are examples of several types of Cartesian to polar equations and their polar transformations, such as straight lines , conic sections, and others.

1.Equation: y=3x+1 (Line)

Step 1: substitute for x and y.

Substitute into the equation y=3x+1

Step 2: Simplify Divide both sides by 𝑟 if 𝑟 ≠ 0 and r = 0.

Step 3: Solve for r

In polar form, the equation  y = 3x+1 becomes:

This equation shows a straight line in polar coordinates.

2. Equation: 𝑥^2 + 𝑦^ 2 = 16 (circle).

Step 1: Substitute for x and y

Substitute into the equation  x ^2 +y^ 2 = 16x:

Step 2: Simplify Divide both sides by  r (assuming 𝑟 ≠0 and r=0):

Polar Form: The equation x^2 + y^2 = 16x in polar form is:

This represents a circle with a radius of 8, centered at (8,0) in Cartesian coordinates.To learn more, check this link.