Vectors - Equation Of Line

Vectors - Equation of a Line

Introduction to vectors
Definition of a line
Parametric equation of a line
Cartesian equation of a line
Equation of a line in three-dimensional space

Vectors - Scalar and Vector Equation of a Line

Scalar equation of a line
Vector equation of a line
Difference between scalar and vector equation of a line
Finding the vector equation of a line given two points
Example: Finding the vector equation of a line passing through two points

Vectors - Direction Ratios and Cosines of a Line

Direction ratios of a line
Direction cosines of a line
Connection between direction ratios and direction cosines
Calculation of direction ratios and direction cosines
Example: Finding direction ratios and direction cosines of a line

Vectors - Collinearity of Two Lines

Collinearity of two lines
Scalar triple product
Relationship between scalar triple product and collinearity
Criteria for collinearity of two lines
Example: Determining if two lines are collinear

Vectors - The Equations of Planes

Definition of a plane
Different forms of the equation of a plane
Cartesian equation of a plane
Normal form of the equation of a plane
Example: Finding the equation of a plane passing through three points

Vectors - Angle between Two Lines

Angle between two lines
Dot product of vectors
Calculation of the angle between two lines using dot product
Angle between two lines in three-dimensional space
Example: Finding the angle between two lines

Vectors - Angle between a Line and a Plane

Angle between a line and a plane
Dot product of vectors
Calculation of the angle between a line and a plane using dot product
Angle between a line and a plane in three-dimensional space
Example: Finding the angle between a line and a plane

Vectors - Intersection of Two Lines

Intersection of two lines
Finding the point of intersection by solving simultaneous equations
Cases of intersection: unique solution, no solution, infinitely many solutions
Example: Finding the point of intersection of two lines

Vectors - Intersection of a Line and a Plane

Intersection of a line and a plane
Finding the point of intersection by solving simultaneous equations
Cases of intersection: unique solution, no solution, line lying on the plane
Example: Finding the point of intersection of a line and a plane

Vectors - Shortest Distance between Two Lines

Shortest distance between two lines
Finding the shortest distance using vector projection
Finding the shortest distance using cross product
Example: Finding the shortest distance between two lines

Vectors - Equation of a Line Introduction:

Vectors represent both magnitude and direction.
In mathematics, lines can also be represented using vectors. Definition of a line:
A line is a straight path that extends infinitely in both directions.
It can be represented by an equation that relates the variables x, y, and z. Parametric equation of a line:
The parametric equation of a line is given by:
- x = x_0 + at
- y = y_0 + bt
- z = z_0 + ct
Where (x_0, y_0, z_0) is a point on the line and a, b, and c are the direction ratios of the line. Cartesian equation of a line:
The Cartesian equation of a line is given by:
- (x - x_0) / a = (y - y_0) / b = (z - z_0) / c
Where (x_0, y_0, z_0) is a point on the line and a, b, and c are the direction ratios of the line. Equation of a line in three-dimensional space:
In three-dimensional space, the equation of a line can be represented using both parametric and Cartesian forms.
The parametric equation represents the line as a set of coordinates, while the Cartesian equation relates the variables directly.

Vectors - Scalar and Vector Equation of a Line Scalar equation of a line:

The scalar equation of a line is given by:
- (x - x_1) / (x_2 - x_1) = (y - y_1) / (y_2 - y_1) = (z - z_1) / (z_2 - z_1)
Where (x_1, y_1, z_1) and (x_2, y_2, z_2) are two different points on the line. Vector equation of a line:
The vector equation of a line is given by:
- r = a + tb
Where r is the position vector of any point on the line, a is the position vector of a specific point on the line, b is the direction vector of the line, and t is a scalar parameter. Difference between scalar and vector equation of a line:
The scalar equation represents a line in terms of ratios of the differences of the variables.
The vector equation represents a line using a position vector, a direction vector, and a scalar parameter. Finding the vector equation of a line given two points:
To find the vector equation of a line given two points (x_1, y_1, z_1) and (x_2, y_2, z_2), we can use the following formula:
- a = (x_1, y_1, z_1)
- b = (x_2 - x_1, y_2 - y_1, z_2 - z_1)
The vector equation of the line is then r = a + tb, where t is a scalar parameter. Example: Finding the vector equation of a line passing through two points:
Consider two points A(1, 2, 3) and B(4, 5, 6).
Using the formula, we find a = (1, 2, 3) and b = (4 - 1, 5 - 2, 6 - 3) = (3, 3, 3).
The vector equation of the line passing through A and B is r = (1, 2, 3) + t(3, 3, 3).

Vectors - Direction Ratios and Cosines of a Line Direction ratios of a line:

Direction ratios of a line are the coefficients of the x, y, and z variables in the vector equation of the line.
For a line with direction vector b = (a, b, c), the direction ratios are a, b, and c. Direction cosines of a line:
Direction cosines of a line are the cosines of the angles that the direction ratios make with the positive x, y, and z axes.
The direction cosines of a line with direction ratios a, b, and c are given by:
- cos(alpha) = a / sqrt(a^2 + b^2 + c^2)
- cos(beta) = b / sqrt(a^2 + b^2 + c^2)
- cos(gamma) = c / sqrt(a^2 + b^2 + c^2) Connection between direction ratios and direction cosines:
The direction cosines can be related to the direction ratios using the following formula:
- a^2 + b^2 + c^2 = cos^2(alpha) + cos^2(beta) + cos^2(gamma) = 1 Calculation of direction ratios and direction cosines:
To calculate the direction ratios of a line with direction vector b = (a, b, c), we simply take the values of a, b, and c.
To calculate the direction cosines, we divide the direction ratios by the magnitude of the direction vector. Example: Finding direction ratios and direction cosines of a line:
Consider a line with direction vector b = (2, -1, 3).
The direction ratios of the line are 2, -1, and 3.
To find the direction cosines, we calculate the magnitude of b:
- |b| = sqrt(2^2 + (-1)^2 + 3^2) = sqrt(14)
The direction cosines are then:
- cos(alpha) = 2 / sqrt(14)
- cos(beta) = -1 / sqrt(14)
- cos(gamma) = 3 / sqrt(14)

Vectors - Collinearity of Two Lines Collinearity of two lines:

Two lines are said to be collinear if they are parallel or coincide with each other.
Collinear lines have the same direction ratios or direction vectors. Scalar triple product:
The scalar triple product of three vectors a, b, and c is defined as:
- a · (b × c)
Where · represents the dot product and × represents the cross product of the vectors. Relationship between scalar triple product and collinearity:
If the scalar triple product of three vectors is zero, the vectors are coplanar and the lines they represent are collinear.
If the scalar triple product is nonzero, the vectors are not coplanar and the lines are not collinear. Criteria for collinearity of two lines:
Two lines are collinear if and only if their direction ratios or direction vectors are proportional.
This can be checked by calculating the scalar triple product of the direction vectors. Example: Determining if two lines are collinear:
Consider two lines with direction vectors a = (2, 1, -3) and b = (-4, -2, 6).
Calculate the scalar triple product: a · (b × b) = (2, 1, -3) · (0, 0, 0) = 0.
Since the scalar triple product is zero, the lines are collinear.

Vectors - The Equations of Planes Definition of a plane:

A plane is a flat surface that extends infinitely in all directions.
It can be represented by an equation that relates the variables x, y, and z. Different forms of the equation of a plane:

Cartesian equation of a plane:
- The cartesian equation of a plane is given by:
  - ax + by + cz + d = 0
- Where a, b, c are the direction ratios of the normal to the plane, and d is a constant.

Normal form of the equation of a plane:
- The normal form of the equation of a plane is given by:
  - (x - x_0)/a = (y - y_0)/b = (z - z_0)/c
- Where (x_0, y_0, z_0) is a point on the plane and a, b, c are the direction ratios of the normal to the plane. Example: Finding the equation of a plane passing through three points:

Consider three points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9).
To find the direction ratios of the normal to the plane, we can calculate the cross product of two vectors in the plane:
- AB = (4-1, 5-2, 6-3) = (3, 3, 3)
- AC = (7-1, 8-2, 9-3) = (6, 6, 6)
- Normal vector = AB × AC = (3, 3, 3) × (6, 6, 6) = (0, 0, 0)
Since the normal vector is zero, the points are collinear and the equation of the plane is undefined.

Vectors - Angle between Two Lines Angle between two lines:

The angle between two lines is defined as the angle formed by their direction vectors. Dot product of vectors:
The dot product of two vectors a = (a_1, a_2, a_3) and b = (b_1, b_2, b_3) is given by:
- a · b = a_1 * b_1 + a_2 * b_2 + a_3 * b_3. Calculation of the angle between two lines using dot product:
The angle theta between two lines with direction vectors a and b is given by:
- cos(theta) = (a · b) / (|a| * |b|)
- theta = arccos[(a · b) / (|a| * |b|)]. Angle between two lines in three-dimensional space:
The angle between two lines is always acute and lies in the range [0, 180°].
If the dot product of the direction vectors is negative, the angle is obtuse. Example: Finding the angle between two lines
Consider two lines with direction vectors a = (1, 2, -3) and b = (4, -5, 6).
Calculate the dot product of the direction vectors: a · b = (1 * 4) + (2 * -5) + (-3 * 6) = -4 - 10 - 18 = -32.
Calculate the magnitudes of the direction vectors: |a| = sqrt(1^2 + 2^2 + (-3)^2) = sqrt(14), |b| = sqrt(4^2 + (-5)^2 + 6^2) = sqrt(77).
Calculate the angle: theta = arccos[(-32) / (sqrt(14) * sqrt(77))].

Vectors - Angle between a Line and a Plane Angle between a line and a plane:

The angle between a line and a plane is defined as the angle formed by the direction vector of the line and the normal vector of the plane. Dot product of vectors:
The dot product of two vectors a = (a_1, a_2, a_3) and b = (b_1, b_2, b_3) is given by:
- a · b = a_1 * b_1 + a_2 * b_2 + a_3 * b_3. Calculation of the angle between a line and a plane using dot product:
The angle theta between a line with direction vector a and a plane with normal vector n is given by:
- cos(theta) = |(a · n)| / (|a| * |n|)
- theta = arccos[|(a · n)| / (|a| * |n|)]. Angle between a line and a plane in three-dimensional space:
The angle between a line and a plane is always acute and lies in the range [0, 180°].
If the dot product of the direction vector and the normal vector is negative, the angle is obtuse. Example: Finding the angle between a line and a plane
Consider a line with direction vector a = (1, 2, -3) and a plane with normal vector n = (4, -5, 6).
Calculate the dot product of the direction vector and the normal vector: a · n = (1 * 4) + (2 * -5) + (-3 * 6) = -3 - 10 - 18 = -31.
Calculate the magnitudes of the direction vector and the normal vector: |a| = sqrt(1^2 + 2^2 + (-3)^2) = sqrt(14), |n| = sqrt(4^2 + (-5)^2 + 6^2) = sqrt(77).
Calculate the angle: theta = arccos[|-31| / (sqrt(14) * sqrt(77))].

Vectors - Intersection of Two Lines Intersection of two lines:

The intersection of two lines occurs when they have a common point or are parallel. Finding the point of intersection by solving simultaneous equations:
To find the point of intersection of two lines, we can set the coordinates of the points on the two lines equal to each other and solve the resulting system of equations. Cases of intersection: unique solution, no solution, infinitely many solutions:

Unique solution:
- If the system of equations has a unique solution, the two lines intersect at a single point.

No solution:
- If the system of equations has no solution, the two lines are parallel and do not intersect.

Infinitely many solutions:
- If the system of equations has infinitely many solutions, the two lines are coincident and overlap with each other. Example: Finding the point of intersection of two lines

Consider two lines with vector equations:
- Line 1: r = (1, 2, 3) + t(2, -1, 3)
- Line 2: r = (3, 4, 5) + s(1, 1, 2)
Set the coordinates equal to each other and form a system of equations:
- 1 + 2t = 3 + s
- 2 + (-t) = 4 + s
- 3 + 3t = 5 + 2s
Solve the system of equations to find the values of t and s.
Substitute the values of t and s back into one of the line equations to find the point of intersection.

Vectors - Intersection of a Line and a Plane Intersection of a line and a plane:

The intersection of a line and a plane occurs when the line lies in the plane or intersects it at a single point. Finding the point of intersection by solving simultaneous equations:
To find the point of intersection of a line and a plane, we can set the coordinates of the points on the line equal to the equation of the plane and solve the resulting system of equations. Cases of intersection: unique solution, no solution, line lying on the plane:

Unique solution:
- If the system of equations has a unique solution, the line intersects the plane at a single point.

No solution:
- If the system of equations has no solution, the line does not intersect the plane.

Line lying on the plane:
- If the system of equations has infinitely many solutions, the line lies on the plane. Example: Finding the point of intersection of a line and a plane

Consider a line with vector equation: r = (1, 2, 3) + t(2, -1, 3)