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Review of Vectors

Vectors Review Test

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The sum of two vectors u and v yields

What is the dot product of two perpendicular vectors?

Which vector operation is most helpful when trying to find the area of a parallelogram?

Which vector operation is most helpful when trying to find the projection of one vector onto another?

The cross product satisfies all of the following criteria EXCEPT

What is v·;(w×v) ?

What is v×(·v)

What happens when a vector is multiplied by a scalar?

What is j×i ?

What is the sum of the vectors that point to the vertices of a cube centered at the origin (in 3-dim space)?

Which of the following strategies is useful in answering the previous question?

Which of the following could not be the sum of two 3-dimensional vectors in the x - y plane?

Which of the following could not be the cross product of two 3- dimensional vectors in the x - y plane?

Which of the following is not a similarity between the dot product and the cross product?

What is (1, 2, 5) + 3(1, 0, - 1) ?

What is (1, 2, 5)·(1, 0, - 1) ?

What is (1, 0, 1)×(0, 14, 0) ?

What is (1, 0, 1)·(0, 14, 0) ?

What is the x -component of the cross product (13, 23, 33)×(- 3, 0, 0) ?

What is (1, 0)×(0, 1) ?

The displacement of a man walking through the desert can be represented by a vector: the magnitude of the vector corresponds to the distance he has walked, while the direction of the vector corresponds to the direction in which he has walked. (This is perhaps the most intuitive example of vectors being applied to real problems). The next 5 questions will deal with such displacement vectors.

If your dog's displacement vector (in a place where your house sits at the origin) is given by (3, 5) , how far is she from home?

If a man walks 3 miles east, then 4 miles north, what is his final displacement vector (in miles)?

If a rabbit makes 3 successive displacements, given by the vectors u , v , and w , what is its total displacement vector from the starting point?

Ernesto Mojito is trying to find his way home late at night. He should have walked in the direction of the unit vector u , but ended up at a displacement v from his original starting point. How far has he gone in the right direction?

Let El Palacio Real be at the origin of Madrid. Assume El Museo del Prado lies (in kilometers) at (4, 2) , and Metropolis lies at (3, 0) . How far must a tourist walk to go from Metropolis to Prado? (Hint: first find the displacement vector using vector subtraction, then compute its magnitude.)

The velocity of a moving car can be represented by a vector: the magnitude of the vector corresponds to the speed of the car, the direction of the vector corresponds to the direction in which the car is moving. If the car speeds up or slows down, the magnitude of its corresponding velocity vector changes (gets longer or shorter). If the car turns, the direction of its velocity vector is also altered (and will rotate to point in the new direction in which the car is heading). This idea will be useful in answering the following 5 questions.

If a car has initial velocity vector v and then doubles in speed, what is the car's new velocity vector?

If a car with initial velocity vector (1, 0) turns 90 degress to the right, without changing its speed, what is its new velocity vector?

What is the speed of a car with velocity (3, 4) ?

A car with initial velocity (3, 4) makes a 37 degree turn to the left, drives 30 minutes in this direction and then turns 85 degrees to the right. After another 20 minutes of travel, the car turns again, and it is now unclear in which direction the car is traveling. However, the car is still traveling at its original speed. Which of the following is a candidate for the present velocity of the car?

A car with initial velocity (1, 0, 0) gets driven off a cliff. Which of the following is a candidate for the car's final velocity as it hits the ocean? (Assume the positive z -direction points upward to the sky).

A line passing through the origin in 3-dimensional space can be characterized by a vector v : all points on the line can be written in the form t v , where t is a real number ( t = 0 yields the origin itself), and the full set of points (ranging over all values of t) is the whole line. For a line that does not pass through the origin, all the points can be written of the form u + t v , where u is a particular vector which can be chosen at will from the points which lie on the line. This idea is central to the following 5 questions.

Which of the following is a set of two distinct parallel lines?

Which of the following is a point on the line given by (1, 2, 1) + t(3, 0, - 1) ?

If one has two lines, given by u 1 + t v 1 and u 2 + t v 2 , which condition ensures that they will be parallel?

If one has two lines, given by u 1 + t v 1 and u 2 + t v 2 , which condition ensures that they will be perpendicular?

In the plane, which line is NOT the same as (1, 0) + t(2, 3) ?

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