This question tests understanding of vectors as quantities that have both magnitude and direction. A vector quantity is characterized by having both a magnitude (size) and a direction, unlike scalar quantities which have only magnitude. Examples of vectors include velocity (speed with direction), force (magnitude with direction of application), and displacement (distance with direction of travel), while scalars include speed (no direction), mass, and temperature. Choice B is correct because it includes both magnitude (400 mph) and direction (25° north of east), fully specifying the vector. Choice D treats the vector as a scalar by omitting direction, but vectors require both magnitude and direction to be fully specified. To distinguish vectors from scalars, ask: does this quantity have a natural direction? Force, velocity, and displacement are vectors; mass, speed, and distance are scalars.

This question tests understanding of finding a vector's component form from its directed line segment representation. If the vector starts at point (x₁, y₁) and ends at point (x₂, y₂), its component form is ⟨x₂ - x₁, y₂ - y₁⟩, found by subtracting initial from terminal point coordinates. For the vector from P(2,-1) to Q(-1,3), we calculate: v = ⟨-1-2, 3-(-1)⟩ = ⟨-3, 4⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point: ⟨-1-2, 3-(-1)⟩ = ⟨-3, 4⟩. Choice C incorrectly reverses the subtraction or makes a sign error, giving ⟨3, -4⟩ instead of ⟨-3, 4⟩, which would represent the opposite direction. When finding vectors from points, be careful with negative coordinates: 3-(-1) = 3+1 = 4, not -4. The negative component -3 indicates leftward movement, while positive 4 indicates upward movement.

This question tests understanding of representing a vector as a directed line segment between two points. If the vector starts at point (x₁, y₁) and ends at point (x₂, y₂), its component form is ⟨x₂ - x₁, y₂ - y₁⟩, found by subtracting initial from terminal point coordinates. For the displacement from A(1,2) to B(4,6), we calculate: d = ⟨4-1, 6-2⟩ = ⟨3, 4⟩. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨4-1, 6-2⟩ = ⟨3, 4⟩. Choice C incorrectly adds the coordinates of the two points instead of finding their difference, giving ⟨5, 8⟩ instead of ⟨3, 4⟩. When finding a vector from two points, always remember: terminal minus initial gives the correct components. The resulting vector ⟨3, 4⟩ tells us the displacement is 3 units right and 4 units up from the starting point.

This question tests understanding of vectors as quantities that have both magnitude and direction. The magnitude of a vector v = ⟨a, b⟩ is calculated using the formula |v| = √(a² + b²), which gives the length of the directed line segment. For vector v = ⟨3, 4⟩, we calculate the magnitude as |v| = √(3² + 4²) = √(9 + 16) = √25 = 5. Choice C is correct because it applies the correct formula with specific values from the stimulus, showing the calculation √(9 + 16) = 5. Choice B incorrectly adds the components instead of adding their squares before taking the square root, giving 3 + 4 = 7 instead of √(9 + 16) = 5. Key to vector problems: remember that magnitude is always calculated as the square root of the sum of squared components, not the sum of the components themselves. For 2D vectors ⟨a, b⟩, think of the components as forming a right triangle: 'a' is the horizontal leg, 'b' is the vertical leg, and the magnitude is the hypotenuse by the Pythagorean theorem.

This question tests understanding of vectors as quantities that have both magnitude and direction. Vectors can be represented in multiple equivalent ways: component form ⟨a, b⟩, as directed line segments with arrows, or in magnitude-direction form. If the vector starts at point (1, 5) and ends at point (-2, 1), its component form is ⟨-2 - 1, 1 - 5⟩ = ⟨-3, -4⟩, found by subtracting initial from terminal point coordinates. Choice A is correct because it applies the correct subtraction for points A(1, 5) to B(-2, 1), giving ⟨-3, -4⟩. Choice B incorrectly reverses the initial and terminal points, which gives the opposite direction and thus the negative of the correct vector. Remember that two vectors are equal if they have the same magnitude and direction, even if they start at different points—position doesn't matter for vector equality. When representing vectors, be consistent with notation: use ⟨a, b⟩ or v⃗ for the vector itself, and |v| or ‖v‖ for its magnitude (a scalar).

This question tests understanding of vectors as quantities that have both magnitude and direction. A directed line segment represents a vector by showing its direction (the arrow) and magnitude (the length of the segment). If the vector starts at point (0, 0) and ends at point (3, 4), its component form is ⟨3, 4⟩, found by subtracting initial from terminal point coordinates. Choice A is correct because from (1,1) to (4,5) gives ⟨4-1, 5-1⟩ = ⟨3,4⟩, matching the magnitude √(9+16)=5 and direction. Choice B incorrectly reverses the initial and terminal points, which gives the opposite direction and thus the negative of the correct vector. Remember that two vectors are equal if they have the same magnitude and direction, even if they start at different points—position doesn't matter for vector equality. When representing vectors, be consistent with notation: use ⟨a, b⟩ or v⃗ for the vector itself, and |v| or ‖v‖ for its magnitude (a scalar).

This question tests understanding of vectors as quantities that have both magnitude and direction. The magnitude of a vector v = ⟨a, b⟩ is calculated using the formula |v| = √(a² + b²), which gives the length of the directed line segment. For vector u = ⟨-6, 8⟩, we calculate the magnitude as |u| = √((-6)² + 8²) = √(36 + 64) = √100 = 10. Choice D is correct because it applies the correct formula with specific values, yielding √(36 + 64) = 10 as the magnitude. Choice C incorrectly adds the components instead of adding their squares before taking the square root, giving |-6| + 8 = 14 instead of √(36 + 64) = 10. Key to vector problems: remember that magnitude is always calculated as the square root of the sum of squared components, not the sum of the components themselves. For 2D vectors ⟨a, b⟩, think of the components as forming a right triangle: 'a' is the horizontal leg, 'b' is the vertical leg, and the magnitude is the hypotenuse by the Pythagorean theorem.

This question tests understanding of vectors as quantities that have both magnitude and direction. Vectors can be represented in multiple equivalent ways: component form ⟨a, b⟩, as directed line segments with arrows, or in magnitude-direction form. The component form ⟨5, 12⟩ means the vector has horizontal component 5 and vertical component 12, which fully determines both its magnitude (√(25 + 144)) and direction (angle arctan(12/5) from positive x-axis). Choice C is correct because |v| is the standard notation for the magnitude, which is a scalar value representing the length. Choice A incorrectly uses v⃗ instead of |v⃗| for magnitude, incorrectly applying vector notation where scalar notation is needed. When representing vectors, be consistent with notation: use ⟨a, b⟩ or v⃗ for the vector itself, and |v| or ‖v‖ for its magnitude (a scalar). Key to vector problems: remember that magnitude is always calculated as the square root of the sum of squared components, not the sum of the components themselves.

This question tests understanding of vectors as quantities that have both magnitude and direction. Vectors can be represented in multiple equivalent ways: component form ⟨a, b⟩, as directed line segments with arrows, or in magnitude-direction form. If the vector starts at point (x₁, y₁) and ends at point (x₂, y₂), its component form is ⟨x₂ - x₁, y₂ - y₁⟩, found by subtracting initial from terminal point coordinates. Choice B is correct because it applies the correct subtraction for points A(2, -1) to B(7, 11), giving ⟨7-2, 11-(-1)⟩ = ⟨5, 12⟩. Choice C incorrectly reverses the initial and terminal points, which gives the opposite direction and thus the negative of the correct vector. Remember that two vectors are equal if they have the same magnitude and direction, even if they start at different points—position doesn't matter for vector equality. When representing vectors, be consistent with notation: use ⟨a, b⟩ or v⃗ for the vector itself, and |v| or ‖v‖ for its magnitude (a scalar).

This question tests understanding of vectors as quantities that have both magnitude and direction. A directed line segment represents a vector by showing its direction (the arrow) and magnitude (the length of the segment). If the vector starts at point (-1, 2) and ends at point (4, 6), its component form is ⟨4 - (-1), 6 - 2⟩ = ⟨5, 4⟩, found by subtracting initial from terminal point coordinates. Choice C is correct because the positive components indicate movement right (positive x) and up (positive y). Choice A incorrectly reverses the initial and terminal points, which gives the opposite direction and thus the negative of the correct vector. Remember that two vectors are equal if they have the same magnitude and direction, even if they start at different points—position doesn't matter for vector equality. When representing vectors, be consistent with notation: use ⟨a, b⟩ or v⃗ for the vector itself, and |v| or ‖v‖ for its magnitude (a scalar).