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

MTH-5110 Pretest Solutions. 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 Solutions 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. C is true. It demonstrates the distributive property of multiplication over addition of vectors.

  2. SCALE 35° • 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. 50 km/h • 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. 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? The resultant components indicates that the vector is in quadrant II. The angle of orientation can be determined using the following formula: A force of 243.8 N in the direction of 134º would have the same effect as the two forces described above.

  4. N swimmer 30° current • 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. Components for swimmer Because the direction of the current is east, there is no vertical component. The 1 km/h is only a horizontal component. The resultant components indicates that the vector is in quadrant III. The angle of orientation can be determined using the following formula: Instead of swimming S 30º W, the current causes the swimmer to swim S 7º W.

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

  6. 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. S 35º E is 270º + 35º = 305º The angle between vectors u and v is: 305º + 170º = 135º

  7. 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. Chasles Relation Distributive Property of multiplication over addition By Hypothesis 1 Chasles Relation Distributive Property of multiplication over addition By Hypothesis 2 Scalar Product of collinear vectors

  8. y B (4,7) A (-5,5) C (11,1) D (2,-1) x STATEMENT TO PROVE: PROOF: Given information Definition of vector addition Definition of vector addition Associative Property for addition of real numbers Definition of vector addition Definition of vector addition Given Information • Prove that quadrilateral ABCD represented below is a rhombus. Quadrilateral ABCD is a rhombus. • Using the vectors below, prove that the addition of vectors is associative.

  9. 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. False True False True False • 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 We know the resultant and are asked to determine the original vector for the ferry. Direction of current 8 km 35º W E CAP-ROUGE Because the direction of the current is west, there is no vertical component. The 3.5 km/h is only a horizontal component but in a negative direction. S The ferry should head out at a speed of 10.41 km/h at an angle of 39º.

  10. 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. The angle between the two vectors is 43º.

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