Vector Concepts for Fluid Mechanics



Vector Concepts for Solid Mechanics

Note: Recall that the equation for a plane is [pic], where [pic], [pic], [pic] and [pic] are constants. If the plane passes through the origin, the [pic]. A vector within that plane, say [pic] must be such that its dot product with the plane’s normal vector [pic] is 0. Thus, if we know the normal vector, we can write [pic], which means that [pic].

1. What is the magnitude of the vector [pic]?

2. If [pic],[pic], and [pic], what is [pic].

3. If [pic], [pic], and [pic] are vectors (you may express answers in dot and cross products):

a. What is the projection of [pic] on [pic]?

b. What is the equation for the plane that is perpendicular to [pic]?

c. What vector is normal to the plane that passes through [pic] and [pic]?

d. What is the equation for the plane that passes through [pic] and [pic]?

e. What is the equation for the line directed along the vector [pic]?

f. What is the equation for the line defined by the intersection of (1) the plane passing through [pic] and [pic] and (2) the plane passing through [pic] and [pic].

g. What is the angle between [pic] and [pic].

h. What is the angle between [pic] and the projection of [pic] on the plane passing through [pic] and [pic].

4. Find a unit vector in the same direction as [pic].

5. Sketch both [pic] and the unit vector in the x, y, z coordinate system.

6. Let [pic] be a vector. What geometric object is described by the equation [pic].

7. Sketch the object in 6 above.

8. If [pic], find the gradient of [pic].

9. Find an equation that describes all of the planes that are perpendicular to the gradient of [pic] at any location in space.

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