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a) Which pair(s) of vectors form opposite vectors? _______________________

MTH-5110 Pretest. 1. Five vectors are represented on the following Cartesian plane. a) Which pair(s) of vectors form opposite vectors? _______________________ b) Which pair(s) of vectors form orthogonal vectors? ______________________

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a) Which pair(s) of vectors form opposite vectors? _______________________

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  1. MTH-5110 Pretest 1. Five vectors are represented on the following Cartesian plane. a) Which pair(s) of vectors form opposite vectors? _______________________ b) Which pair(s) of vectors form orthogonal vectors? ______________________ c) Name the vector(s) that could form a vector basis with vector w. ___________ 2. From the three expressions given below, determine which one is true. State the property associated with the true expression.

  2. SCALE • A small plane flies in a direction N 35° W at 250 km/h. A wind blows to the south at 55 km/h to change its direction. Construct a vector diagram to show the resultant. • Given the following ordered pairs: A (-1, 3), B (3.5, 4), C (5, -2) and D (1, -5), use the algebraic method to calculate the components of the resultant vector, r.

  3. N swimmer 30° current • Two forces, one 150 N and the other 220 N are applied to the same object. The first force is oriented SW and the second at 125°. What single force would have the same effect on this object? • A swimmer crosses a river in a direction S 30° W at a speed of 2.5 km/h. A current of 1 km/h towards the east changes the swimmer’s direction. What is the angle of deviation from the intended direction.

  4. Calculate the components of the resultant vector w. • Determine coefficients k1 and k2 of this linear combination.

  5. 190 N 43° 150 m • Given vectors u and v below: • Solve the following problems. • A mother pulls her child in a sleigh. The rope on the sleigh makes a 43° angle with the horizontal and she pulls with a force of 190 N to cause it to move. How much work (in kilojoules) is done to make the sleigh move 150 m.

  6. B Scalar product of orthogonal vectors Definition of an altitude A C P Definition of a norm of a vector • Using vector theory, complete the following proof. • In a right triangle, the measure of a side adjacent to the right angle is the geometric mean of its projection and the hypotenuse. Approach: We can apply the definition of a norm of a vector to side AB and rewrite side AB as the sum of vectors using Chasles relation. The expression can be transformed to eventually reach the sired conclusion.

  7. y B (4,7) A (-5,5) C (11,1) D (2,-1) x • Prove that quadrilateral ABCD represented below is a rhombus. • Using the vectors below, prove that the addition of vectors is associative.

  8. B A C D • Triangle ABC is a right triangle whose altitude is segment AD. Determine whether the statements concerning this figure are true or false. Adjust the false statements to make them true. • A ferry is used to transport passenger vehicles between Cap-Rouge and Cap-Espoir, a distance of 8 km. Its direction with respect to Cap-Rouge is N 35º E. The current of the river that the ferry crosses is 3.5 km/h to the west. So as to land at the Cap-Espoir in one hour, at what speed and in what direction should the captain set out in the ferry? CAP-ESPOIR N Direction of current 8 km 35º W E CAP-ROUGE S

  9. Given the following ordered pairs: A (-5.2, -3), B (-3, 2.5), C (-1, 7) and D (3.3, -4), Using the scalar product, calculate the angle between vectors s and t.

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