Difference between revisions of "ModelCourses/Calculus/Vectors/Vectors in Space"
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* Vector Algebra |
* Vector Algebra |
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| − | ** Expressing a vector from Point A to Point B in vector notation |
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| + | ** The right-handed coordinate system, three axes, three coordinate planes and eight octants |
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| − | ** Sketching a position vector |
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| + | *** Sketch a point in space. |
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| − | ** Vector algebra: (1) scalar multiplication; (2) vector addition and substraction |
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| + | *** Sketch a line that passes through a given point and is parallel to an axis. |
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| − | ** Magnitude and 2-norm of a vector, unit vector |
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| + | *** Sketch a plane that contains a point and is parallel to a coordinate plane. |
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| + | *** Sketch a plane that contains a point and is perpendicular to an axis. |
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| + | *** Express a vector from Point A to Point B in vector notation. |
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| + | *** Sketch a position vector. |
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| + | ** Vector algebra: (1) scalar multiplication; (2) vector addition and subtraction |
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| + | *** Compute and sketching a scalar times a vector and a sum (difference) of two vectors. |
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** Triangle inequality |
** Triangle inequality |
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| − | * |
+ | * Dot Product and Applications |
** Two definitions of dot product of two vectors |
** Two definitions of dot product of two vectors |
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| − | *** Calculating the standard collection of numerical examples |
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| + | ** Angle between two vectors |
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| − | ** Parallel and geometric implications |
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| + | *** Compute the dot product of two vectors. |
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| − | *** Given a particular vector, create other parallel vectors of desired length |
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| + | *** Compute the angle between two vectors. |
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| − | ** Angle between vectors, orthogonality and cos(theta) |
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| + | *** Determine if two vectors are parallel or orthogonal (perpendicular) when the cosine of the angle between these two vector is 1, -1, or 0. |
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| − | *** Given two vectors, determine the angle between |
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| + | *** Determine if the angle between two vectors is acute or obtuse when the dot product of these two vectors is positive or negative. |
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| − | *** Given one vector and an angle, determine another other vector with the desired angle. Maybe give part of the second vector. |
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| + | *** Create a vector v that is parallel to a given vector. |
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| − | *** |
+ | *** Create a vector v that is orthogonal to a given vector. |
| + | *** Given a vector u and an angle theta, create a vector v such that the angle between u and v is theta. |
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| + | ** Projection and component of vector u onto vector v |
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| + | *** Compute the work done by a force vector along a direction vector. |
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| + | *** Compute the distance from a given point to a given line. |
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| + | *** Compute the distance between two planes. |
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| − | * |
+ | * Cross Product and Applications |
| − | ** |
+ | ** Definition of the cross product of two vectors in space |
| − | ** |
+ | ** The cross product of vectors u and v is orthogonal (perpendicular) to u and v and satisfies the right-handed rule. |
| − | *** Given |
+ | *** Given two vectors u and v that are not parallel, find a vector which is orthogonal to both u and v. |
| − | ** |
+ | *** Compute the area of the parallelogram whose two sides are formed by two given vectors. |
| − | *** |
+ | *** Compute the volume of the parallelepiped whose three sides are formed by three given vectors. |
[[ModelCourses/Calculus/Vectors/setUnit1|Download the set definition file for this problem set]] |
[[ModelCourses/Calculus/Vectors/setUnit1|Download the set definition file for this problem set]] |
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[[ModelCourses/Multivariate Calculus]] |
[[ModelCourses/Multivariate Calculus]] |
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| + | |||
| + | [[Category:Model_Courses]] |
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Latest revision as of 09:21, 22 June 2021
Vectors in Space
- Vector Algebra
- The right-handed coordinate system, three axes, three coordinate planes and eight octants
- Sketch a point in space.
- Sketch a line that passes through a given point and is parallel to an axis.
- Sketch a plane that contains a point and is parallel to a coordinate plane.
- Sketch a plane that contains a point and is perpendicular to an axis.
- Express a vector from Point A to Point B in vector notation.
- Sketch a position vector.
- Vector algebra: (1) scalar multiplication; (2) vector addition and subtraction
- Compute and sketching a scalar times a vector and a sum (difference) of two vectors.
- Triangle inequality
- The right-handed coordinate system, three axes, three coordinate planes and eight octants
- Dot Product and Applications
- Two definitions of dot product of two vectors
- Angle between two vectors
- Compute the dot product of two vectors.
- Compute the angle between two vectors.
- Determine if two vectors are parallel or orthogonal (perpendicular) when the cosine of the angle between these two vector is 1, -1, or 0.
- Determine if the angle between two vectors is acute or obtuse when the dot product of these two vectors is positive or negative.
- Create a vector v that is parallel to a given vector.
- Create a vector v that is orthogonal to a given vector.
- Given a vector u and an angle theta, create a vector v such that the angle between u and v is theta.
- Projection and component of vector u onto vector v
- Compute the work done by a force vector along a direction vector.
- Compute the distance from a given point to a given line.
- Compute the distance between two planes.
- Cross Product and Applications
- Definition of the cross product of two vectors in space
- The cross product of vectors u and v is orthogonal (perpendicular) to u and v and satisfies the right-handed rule.
- Given two vectors u and v that are not parallel, find a vector which is orthogonal to both u and v.
- Compute the area of the parallelogram whose two sides are formed by two given vectors.
- Compute the volume of the parallelepiped whose three sides are formed by three given vectors.
