FAQs on Cartesian Coordinates in 3 Dimensions Video Lecture  Mathematics (Maths) Class 11  Commerce
1. What are Cartesian coordinates in 3 dimensions? 

Ans. Cartesian coordinates in 3 dimensions refer to a coordinate system that uses three perpendicular axes, namely the xaxis, yaxis, and zaxis, to specify the position of a point in threedimensional space. Each point is represented by an ordered triple (x, y, z), where x represents the distance from the origin along the xaxis, y represents the distance from the origin along the yaxis, and z represents the distance from the origin along the zaxis.
2. How do you find the distance between two points in 3D Cartesian coordinates? 

Ans. To find the distance between two points in 3D Cartesian coordinates, you can use the distance formula derived from the Pythagorean theorem. Let's say you have two points A(x1, y1, z1) and B(x2, y2, z2). The distance between these two points can be calculated using the formula:
distance = sqrt((x2  x1)^2 + (y2  y1)^2 + (z2  z1)^2)
This formula calculates the square root of the sum of the squares of the differences in the x, y, and z coordinates of the two points.
3. How do you calculate the midpoint of a line segment in 3D Cartesian coordinates? 

Ans. To find the midpoint of a line segment in 3D Cartesian coordinates, you can use the midpoint formula. Suppose you have two points A(x1, y1, z1) and B(x2, y2, z2) that define the endpoints of the line segment. The midpoint M(x, y, z) can be calculated using the following formulas:
x = (x1 + x2) / 2
y = (y1 + y2) / 2
z = (z1 + z2) / 2
These formulas calculate the average of the x, y, and z coordinates of the two endpoints to determine the coordinates of the midpoint.
4. How do you determine the equation of a plane in 3D Cartesian coordinates? 

Ans. To determine the equation of a plane in 3D Cartesian coordinates, you need to know a point on the plane and the normal vector of the plane. Let's say you have a point P(x1, y1, z1) on the plane and a normal vector N(a, b, c). The equation of the plane can be written in the form:
ax + by + cz = d
To find the value of d, you can substitute the coordinates of the point P into the equation:
d = ax1 + by1 + cz1
Thus, the equation of the plane in 3D Cartesian coordinates becomes ax + by + cz = ax1 + by1 + cz1.
5. How do you calculate the angle between two vectors in 3D Cartesian coordinates? 

Ans. To calculate the angle between two vectors in 3D Cartesian coordinates, you can use the dot product formula. Let's say you have two vectors A(x1, y1, z1) and B(x2, y2, z2). The dot product of these two vectors can be calculated as:
A · B = (x1 * x2) + (y1 * y2) + (z1 * z2)
The angle θ between these two vectors can be determined using the formula:
θ = acos((A · B) / (A * B))
Here, A and B represent the magnitudes (lengths) of vectors A and B, respectively. The acos function returns the inverse cosine of the dot product divided by the product of the magnitudes, yielding the angle between the two vectors in radians.