Calculus and Vectors - Lines and Planes

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Start date Apr 6, 2020
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Calculus Lines Planes Vectors

In summary, lines and planes are one-dimensional and two-dimensional objects, respectively, used to represent motion and position in calculus and vectors. To find the equation of a line given two points, you can use the point-slope or slope-intercept form. Vectors are often used to represent the direction and magnitude of lines and planes, and the equation of a line/plane can be written in vector form. Two lines are parallel if they have the same slope and perpendicular if their slopes are negative reciprocals. The distance between a point and a line/plane can be found using a formula derived from the distance formula.

Apr 6, 2020

ttpp1124

Homework Statement: The planes A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 are perpendicular. Find the value of A1A2 + B1B2 + C1C2.

I solved it, can someone confirm? Thanks!

Relevant Equations: n/a

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Apr 6, 2020

Mark44

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ttpp1124 said:

Homework Statement:: The planes A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 are perpendicular. Find the value of A1A2 + B1B2 + C1C2.

I solved it, can someone confirm? Thanks!
Relevant Equations:: n/a

View attachment 260145

Yes. Clearly ##A_1A_2 + B_1B_2 + C_1C_2## is the dot product of the two normals to the planes, so this expression will be zero.

Last edited: Apr 6, 2020

1. What is the difference between a line and a plane in calculus and vectors?

A line is a one-dimensional object that extends infinitely in both directions, while a plane is a two-dimensional object that extends infinitely in all directions. In calculus and vectors, lines are typically represented by equations in the form of y = mx + b, while planes are represented by equations in the form of ax + by + cz = d.

2. How do you find the equation of a line or plane in calculus and vectors?

To find the equation of a line, you need to know at least two points on the line and use the slope-intercept formula (y = mx + b) to solve for the slope (m) and y-intercept (b). To find the equation of a plane, you need to know at least three points on the plane and use the general equation (ax + by + cz = d) to solve for the coefficients (a, b, and c) and the constant (d).

3. What is the significance of the direction vector in lines and planes in calculus and vectors?

The direction vector represents the direction in which the line or plane is oriented. In lines, the direction vector is the slope (m) of the line, while in planes, the direction vector is the normal vector (a, b, c) perpendicular to the plane.

4. How do you determine if two lines or planes are parallel or perpendicular in calculus and vectors?

To determine if two lines are parallel, you can compare their slopes. If the slopes are equal, the lines are parallel. To determine if two planes are parallel, you can compare their normal vectors. If the normal vectors are parallel, the planes are parallel. To determine if two lines or planes are perpendicular, you can take the dot product of their direction vectors. If the dot product is equal to zero, the lines or planes are perpendicular.

5. How can calculus and vectors be applied to real-world problems involving lines and planes?

Calculus and vectors can be used to model and analyze real-world situations involving lines and planes, such as the motion of objects in space or the flow of fluids through pipes. They can also be used to optimize and solve problems related to lines and planes, such as finding the shortest distance between two points or the maximum volume of a container with a given surface area.