This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude of the resultant. To add vectors given in magnitude and direction form, first convert each vector to component form: a vector with magnitude r and direction θ (measured counterclockwise from the positive x-axis) has components ⟨r·cos(θ), r·sin(θ)⟩, then add the components of all vectors, and finally compute the magnitude and direction of the resulting component form. Vector u has magnitude 10 at direction 30°, so its components are ⟨10·cos(30°), 10·sin(30°)⟩ = ⟨10·(√3/2), 10·(1/2)⟩ = ⟨5√3, 5⟩. Vector v has magnitude 10 at direction 150°, so its components are ⟨10·cos(150°), 10·sin(150°)⟩ = ⟨10·(-√3/2), 10·(1/2)⟩ = ⟨-5√3, 5⟩. Adding these gives u + v = ⟨5√3 + (-5√3), 5 + 5⟩ = ⟨0, 10⟩, and the magnitude is √(0² + 10²) = √100 = 10. Choice A is correct because it properly converts to components, adds correctly, and applies the Pythagorean theorem for magnitude. Choice B incorrectly adds the magnitudes directly (10 + 10 = 20), but vector addition requires converting to components first—you can only add magnitudes directly when vectors point in the same direction. Key to adding vectors in magnitude-direction form: always convert to components first using x = r·cos(θ) and y = r·sin(θ), add the x-components together and y-components together, then find magnitude using √(x² + y²). To verify your answer, check that the magnitude of the sum is between |r₁ - r₂| and r₁ + r₂ (triangle inequality), and that the direction makes geometric sense given the original vectors' directions.

This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude and direction of the resultant. To add vectors given in magnitude and direction form, first convert each vector to component form: a vector with magnitude r and direction θ (measured counterclockwise from the positive x-axis) has components ⟨r·cos(θ), r·sin(θ)⟩, then add the components of all vectors, and finally compute the magnitude and direction of the resulting component form. Since the hiker walks 6 km east and 8 km north, these are perpendicular displacements forming a right triangle. The magnitude of the resultant displacement is the hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 km. The direction angle is arctan(8/6) = arctan(4/3), measured from the positive x-axis (east direction). Choice A is correct because it properly identifies both the magnitude as 10 km and the direction as tan⁻¹(8/6), which correctly represents the angle whose tangent is the ratio of northward to eastward displacement. Choice C incorrectly gives the direction as tan⁻¹(6/8), which inverts the ratio—the correct ratio for direction from east is (north component)/(east component) = 8/6, not 6/8. Special case shortcut: when vectors are perpendicular (like one pointing east and one pointing north), you can directly apply the Pythagorean theorem to the magnitudes: |u + v| = √(r₁² + r₂²), and the direction is simply arctan(r₂/r₁) from the first vector's direction. This is a classic 6-8-10 right triangle, making the calculation particularly clean.

This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude and direction of the resultant. For special cases like perpendicular vectors, the calculation simplifies: if one vector is magnitude r₁ along the x-axis (direction 0°) and another is magnitude r₂ along the y-axis (direction 90°), the resultant magnitude is simply √(r₁² + r₂²) and the direction is arctan(r₂/r₁). Since one vector points east with magnitude 5 and the other points north with magnitude 12, these are perpendicular, forming a right triangle. The magnitude of the sum is the hypotenuse: √(5² + 12²) = √(25 + 144) = √169 =13, and the direction is arctan(12/5) from the positive x-axis. Choice A is correct because it properly converts to components, adds correctly, and applies the Pythagorean theorem for magnitude and correctly uses arctan with the proper ratio for direction. Choice D incorrectly switches the ratio in the arctan, calculating arctan(5/12) instead of arctan(12/5), perhaps confusing the magnitudes of the vectors. Key to adding vectors in magnitude-direction form: always convert to components first using x = r·cos(θ) and y = r·sin(θ), add the x-components together and y-components together, then find magnitude using √(x² + y²) and direction using arctan(y/x) with quadrant adjustment. Special case shortcut: when vectors are perpendicular (like one pointing east and one pointing north), you can directly apply the Pythagorean theorem to the magnitudes: |u + v| = √(r₁² + r₂²), and the direction is arctan(r₂/r₁) from the first vector's direction.

This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude of the resultant. To add vectors given in magnitude and direction form, first convert each vector to component form: a vector with magnitude r and direction θ (measured counterclockwise from the positive x-axis) has components $\langle r \cdot \cos(\theta), r \cdot \sin(\theta) \rangle$, then add the components of all vectors, and finally compute the magnitude of the resulting component form. Vector u has magnitude 5 at direction 60°, so its components are $\langle 5 \cdot \cos(60^\circ), 5 \cdot \sin(60^\circ) \rangle = \langle 2.5, \frac{5\sqrt{3}}{2} \rangle$. Vector v has magnitude 5 at direction 240°, so its components are $\langle 5 \cdot \cos(240^\circ), 5 \cdot \sin(240^\circ) \rangle = \langle -2.5, -\frac{5\sqrt{3}}{2} \rangle$. Adding these gives u + v = $\langle 0, 0 \rangle$, and the magnitude is $\sqrt{0 + 0} = 0$. Choice C is correct because it properly converts to components, adds correctly, and applies the Pythagorean theorem for magnitude. Choice A incorrectly adds the magnitudes directly (5 + 5 =10), but vector addition requires converting to components first—you can only add magnitudes directly when vectors point in the same direction. Don't fall for the trap of adding magnitudes directly—this only works when vectors point in exactly the same direction; otherwise, components partially cancel and you must use the component method.

This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude of the resultant. For special cases like perpendicular vectors, the calculation simplifies: if one vector is magnitude r₁ along the x-axis (direction 0°) and another is magnitude r₂ along the y-axis (direction 90°), the resultant magnitude is simply √(r₁² + r₂²). Since one vector points east with magnitude 8 and the other points north with magnitude 15, these are perpendicular, forming a right triangle. The magnitude of the sum is the hypotenuse: √(8² + 15²) = √(64 + 225) = √289 =17. Choice B is correct because it properly applies the Pythagorean theorem for magnitude. Choice A incorrectly adds the magnitudes directly (8 + 15 = 23), but vector addition requires converting to components first—you can only add magnitudes directly when vectors point in the same direction. Special case shortcut: when vectors are perpendicular (like one pointing east and one pointing north), you can directly apply the Pythagorean theorem to the magnitudes: |u + v| = √(r₁² + r₂²), and the direction is arctan(r₂/r₁) from the first vector's direction. Don't fall for the trap of adding magnitudes directly—this only works when vectors point in exactly the same direction; otherwise, components partially cancel and you must use the component method.

This question tests understanding of how to add vectors given in magnitude and direction form and find the direction of the resultant. The direction of a vector with components ⟨a, b⟩ is found using θ = arctan(b/a), but you must check which quadrant the vector is in based on the signs of a and b: Quadrant I (both positive) gives 0° to 90°, Quadrant II (a negative, b positive) gives 90° to 180°, Quadrant III (both negative) gives 180° to 270°, and Quadrant IV (a positive, b negative) gives 270° to 360°. Vector u has magnitude 6 at direction 45°, so its components are ⟨3√2, 3√2⟩. Vector v has magnitude 6 at direction 315°, so its components are ⟨3√2, -3√2⟩. Adding these gives u + v = ⟨6√2, 0⟩. The direction angle is arctan(0/(6√2)) = 0°. Choice A is correct because it correctly uses arctan with quadrant adjustment for direction. Choice D gives the direction of only one of the original vectors instead of the direction of their sum.

This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude and direction of the resultant. To add vectors given in magnitude and direction form, first convert each vector to component form: a vector with magnitude r and direction θ (measured counterclockwise from the positive x-axis) has components ⟨r·cos(θ), r·sin(θ)⟩, then add the components of all vectors, and finally compute the magnitude and direction of the resulting component form. Vector u has magnitude 10 at direction 0°, so its components are ⟨10, 0⟩. Vector v has magnitude 6 at direction 180°, so its components are ⟨-6, 0⟩. Adding these gives u + v = ⟨4, 0⟩, the magnitude is √(16 + 0) =4, and the direction is 0° since it's along the positive x-axis. Choice A is correct because it properly converts to components, adds correctly, and determines both magnitude and direction accurately. Choice C makes an arithmetic error in the calculation, computing 10 - 6 =4 but placing it at 180° instead of 0°, confusing the direction of the resultant. Key to adding vectors in magnitude-direction form: always convert to components first using x = r·cos(θ) and y = r·sin(θ), add the x-components together and y-components together, then find magnitude using √(x² + y²) and direction using arctan(y/x) with quadrant adjustment.

This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude of the resultant. To add vectors given in magnitude and direction form, first convert each vector to component form: a vector with magnitude r and direction θ (measured counterclockwise from the positive x-axis) has components ⟨r·cos(θ), r·sin(θ)⟩, then add the components of all vectors, and finally compute the magnitude of the resulting component form. Vector u has magnitude 10 at direction 30°, so its components are ⟨10·cos(30°), 10·sin(30°)⟩ = ⟨5√3, 5⟩. Vector v has magnitude 10 at direction 150°, so its components are ⟨10·cos(150°), 10·sin(150°)⟩ = ⟨-5√3, 5⟩. Adding these gives u + v = ⟨0, 10⟩, and the magnitude is √(0² + 10²) = 10. Choice A is correct because it properly converts to components, adds correctly, and applies the Pythagorean theorem for magnitude. Choice B incorrectly adds the magnitudes directly (10 + 10 = 20), but vector addition requires converting to components first—you can only add magnitudes directly when vectors point in the same direction. Don't fall for the trap of adding magnitudes directly—this only works when vectors point in exactly the same direction; otherwise, components partially cancel and you must use the component method.

This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude and direction of the resultant. For special cases like perpendicular vectors, the calculation simplifies: if one vector is magnitude r₁ along the x-axis (direction 0°) and another is magnitude r₂ along the y-axis (direction 90°), the resultant magnitude is simply √(r₁² + r₂²) and the direction is arctan(r₂/r₁). Since the vectors have equal magnitude 13 and are perpendicular, forming a right triangle. The magnitude of the sum is √(13² + 13²) = √(169 + 169) = √338 =13√2, and the direction is arctan(13/13)=45°. Choice A is correct because it properly converts to components, adds correctly, and applies the Pythagorean theorem for magnitude and correctly uses arctan for direction. Choice B incorrectly adds the magnitudes directly (13 + 13 =26), but vector addition requires converting to components first—you can only add magnitudes directly when vectors point in the same direction. Key to adding vectors in magnitude-direction form: always convert to components first using x = r·cos(θ) and y = r·sin(θ), add the x-components together and y-components together, then find magnitude using √(x² + y²) and direction using arctan(y/x) with quadrant adjustment. Special case shortcut: when vectors are perpendicular (like one pointing east and one pointing north), you can directly apply the Pythagorean theorem to the magnitudes: |u + v| = √(r₁² + r₂²), and the direction is arctan(r₂/r₁) from the first vector's direction.

This question tests understanding of how to add vectors given in magnitude and direction form and find the magnitude of the resultant. The magnitude of a vector sum is found using the Pythagorean theorem: after converting each vector to components and adding, if the sum is ⟨a, b⟩, then the magnitude is |u + v| = √(a² + b²). Vector u has magnitude 5 at direction 0°, so its components are ⟨5·cos(0°), 5·sin(0°)⟩ = ⟨5, 0⟩; vector v has magnitude 12 at direction 90°, so its components are ⟨12·cos(90°), 12·sin(90°)⟩ = ⟨0, 12⟩; adding these gives u + v = ⟨5, 12⟩, and the magnitude is √(5² + 12²) = √(25 + 144) = √169 = 13. Choice B is correct because it properly converts to components, adds correctly, and applies the Pythagorean theorem for magnitude. Choice A incorrectly adds the magnitudes directly (5 + 12 = 17), but vector addition requires converting to components first—you can only add magnitudes directly when vectors point in the same direction. Key to adding vectors in magnitude-direction form: always convert to components first using x = r·cos(θ) and y = r·sin(θ), add the x-components together and y-components together, then find magnitude using √(x² + y²) and direction using arctan(y/x) with quadrant adjustment. Special case shortcut: when vectors are perpendicular (like one pointing east and one pointing north), you can directly apply the Pythagorean theorem to the magnitudes: |u + v| = √(r₁² + r₂²), and the direction is simply arctan(r₂/r₁) from the first vector's direction.