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Work, vectors, and the inner product

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4. A 1.5 kg object moving along the x axis has a velocity of +4.0 m/s at x = 0. If the only force acting on this object is shown in the figure, what is the kinetic energy of the object at x = +3.0m?

5. If the vectors A and B have magnitudes of 10 and 11, respectively, and the scalar product of these two vectors is -100, what is the magnitude of the sum of these two vectors?

6. Two vectors A and B are given by A = 4i + 8j and B = 6i - 2j. The scalar product of A and third vector C is -16. The scalar product of B and C is +18. The z component of C is 0. What is the magnitude of C?

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Problem 4

Let's first compute how much work the force performs on the object as it moves from x = 0 meters to x = 3 meters. This is given by the integral of F dx. From x = 0 meters to 2 meters F is positive and the contribution to the integral is the area enclosed by the x-axis, the y-axis and the graph of the force which is 8 J.

From x = 2 meters to 3 meters the contribution to the integral is minus the area anclosed by the graph, the segment on the x-axis from x = 2 meters to x = 3 meters and the line parallel to the y axis given by x = 3 meters. This contribution is thus minus 2 J.

So, the total gain in kinetic energy equals 8 J - 2 J = 6 J. The kinetic energy at x = 0 meters was

1/2 * 1.5 kg (4.0 m/s)^2 = 12 J

So, it follows that the kinetic energy at x = 3 meters is 12 J + 6 J = 18 J which is answer a)

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Solution Summary

The following post helps with problems involving kinetic energy, magnitude and vectors.

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Linear Algebra: Vector Space Problems

Please see attachment.

The set of elements belonging to R^2 is usually denoted as {(a, b) | a, b ∈ R}. Combining elements within this set under the operations of addition and scalar multiplication should use the following notation:
Addition Example: (-2, 10) + (-5, 0) = (-2 - 5, 10 + 0) = (-7, 10)
Scalar Multiplication Example: -10 × (1, -7) = (-10 × 1, -10 × -7) = (-10, 70), where -10 is a scalar.
Write an explanation of vector space where you:
1) Provide a mathematical definition for a vector space.
2) Indicate whether R^2 is a vector space.
* Justify assertions by applying the provided mathematical definition for a vector space.
3) Determine whether R^2 is spanned by (1, 1) and (3, 2) (show all work).
4) Define a nontrivial subspace of R^2 (show all work).

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