This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. When a vector v is multiplied by a scalar c, the magnitude of the result is |cv| = |c|·|v| (the absolute value of c times the magnitude of v), and the direction either stays the same (if c > 0) or reverses 180° (if c < 0). For scalar c = -1/2, the magnitude halves because |c| = 1/2, giving |cv| = (1/2)·|v|, and the direction reverses because c is negative. Choice B is correct because it correctly states both magnitude and direction. Choice A incorrectly claims the direction stays the same when c = -1/2, but since c is negative, the direction actually reverses 180°. Special scalars to remember: c = 1 (no change), c = -1 (flip direction only), c = 2 (double length, same direction), c = -2 (double length, opposite direction), c = 1/2 (half length, same direction). For direction: think of the sign of c as a switch—positive means 'keep the same direction,' negative means 'flip 180° to the opposite direction,' and the magnitude of c only affects how much to scale, not which way to point.

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. When a vector v is multiplied by a scalar c, the magnitude of the result is |cv| = |c|·|v| (the absolute value of c times the magnitude of v), and the direction either stays the same (if c > 0) or reverses 180° (if c < 0). The magnitude is |cv| = |3|·|v| = 3·12 = 36, and since c is positive, the direction remains the same as the original vector v's direction of 30°. Choice A is correct because it properly applies |cv| = |c|·|v| and correctly identifies the direction based on the sign of c. Choice B incorrectly claims the direction reverses when c = 3, but since c is positive, the direction actually stays the same. Key to scalar multiplication: magnitude always scales by |c| (the absolute value), so |cv| = |c|·|v|, while direction depends on the sign of c—positive preserves direction, negative reverses it. Special scalars to remember: c = 1 (no change), c = -1 (flip direction only), c = 2 (double length, same direction), c = -2 (double length, opposite direction), c = 1/2 (half length, same direction).

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. The formula |cv| = |c|·|v| tells us that the magnitude scales by the absolute value of the scalar: |c| > 1 stretches the vector, 0 < |c| < 1 compresses it, and the absolute value ensures the magnitude is always positive regardless of whether c is positive or negative. Given |v| = √(9 + 16) = √25 = 5 and scalar c = 2, we apply the formula: |cv| = |2|·5 = 2·5 = 10. Choice C is correct because it properly applies |cv| = |c|·|v|. Choice D makes an arithmetic error, calculating 4·5 = 20 instead of 2·5 = 10. When computing from components v = ⟨a, b⟩, remember cv = ⟨ca, cb⟩, and then find magnitude using |cv| = √((ca)² + (cb)²) = |c|√(a² + b²), confirming the formula. Key to scalar multiplication: magnitude always scales by |c| (the absolute value), so |cv| = |c|·|v|, while direction depends on the sign of c—positive preserves direction, negative reverses it.

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. When a vector v is multiplied by a scalar c, the magnitude of the result is |cv| = |c|·|v| (the absolute value of c times the magnitude of v), and the direction either stays the same (if c > 0) or reverses 180° (if c < 0). The magnitude is |(1/3)v| = |1/3|·18 = (1/3)·18 = 6 N, and since c is positive, the direction remains the same as the original vector v's direction of north. Choice A is correct because it properly applies |cv| = |c|·|v| to get magnitude 6 and correctly identifies the direction remains the same based on the positive sign of c. Choice B incorrectly claims the direction reverses to south when c = 1/3, but since c is positive, the direction actually stays the same. Key to scalar multiplication: magnitude always scales by |c| (the absolute value), so |cv| = |c|·|v|, while direction depends on the sign of c—positive preserves direction, negative reverses it. Special scalars to remember: c = 1 (no change), c = -1 (flip direction only), c = 2 (double length, same direction), c = -2 (double length, opposite direction), c = 1/2 (half length, same direction).

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. When a vector v is multiplied by a scalar c, the magnitude of the result is |cv| = |c|·|v| (the absolute value of c times the magnitude of v), and the direction either stays the same (if c > 0) or reverses 180° (if c < 0). The magnitude is |cv| = |-3|·50 = 3·50 = 150, and since c is negative, the direction is opposite to the original vector v's direction of east, so west. Choice A is correct because it properly applies |cv| = |c|·|v| and correctly identifies direction based on sign of c. Choice C forgets to take the absolute value of c, computing |cv| = c·|v| = -150, but magnitude must always be positive (|cv| = |c|·|v|). Key to scalar multiplication: magnitude always scales by |c| (the absolute value), so |cv| = |c|·|v|, while direction depends on the sign of c—positive preserves direction, negative reverses it. For direction: think of the sign of c as a switch—positive means 'keep the same direction,' negative means 'flip 180° to the opposite direction,' and the magnitude of c only affects how much to scale, not which way to point.

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. For direction, the sign of c determines the result: positive scalars preserve the direction of the original vector, while negative scalars reverse it by 180°, making cv point in exactly the opposite direction from v. Since the scalar c = -1 is negative, the direction of cv is opposite to v (reversed 180°). Specifically, if v points at 60°, then cv points at 60° + 180° = 240°. Choice C is correct because it correctly identifies direction based on sign of c. Choice A incorrectly claims the direction stays the same when c = -1, but since c is negative, the direction actually reverses 180°. For direction: think of the sign of c as a switch—positive means 'keep the same direction,' negative means 'flip 180° to the opposite direction,' and the magnitude of c only affects how much to scale, not which way to point. Key to scalar multiplication: magnitude always scales by |c| (the absolute value), so |cv| = |c|·|v|, while direction depends on the sign of c—positive preserves direction, negative reverses it.

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. The formula |cv| = |c|·|v| tells us that the magnitude scales by the absolute value of the scalar: |c| > 1 stretches the vector, 0 < |c| < 1 compresses it, and the absolute value ensures the magnitude is always positive regardless of whether c is positive or negative. Given |v| = 12 and scalar c = 1/2, we apply the formula: |cv| = |1/2|·12 = (1/2)·12 = 6, and since c is positive, the direction remains the same as v at 45°. Choice A is correct because it properly applies |cv| = |c|·|v| and correctly identifies direction based on sign of c. Choice C incorrectly claims the direction reverses to 225° when c = 1/2, but since c is positive, the direction actually stays the same. For direction: think of the sign of c as a switch—positive means 'keep the same direction,' negative means 'flip 180° to the opposite direction,' and the magnitude of c only affects how much to scale, not which way to point. Special scalars to remember: c = 1 (no change), c = -1 (flip direction only), c = 2 (double length, same direction), c = -2 (double length, opposite direction), c = 1/2 (half length, same direction).

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. When a vector v is multiplied by a scalar c, the magnitude of the result is |cv| = |c|·|v| (the absolute value of c times the magnitude of v), and the direction either stays the same (if c > 0) or reverses 180° (if c < 0). Given |v| = √(64 + 225) = √289 = 17 and scalar c = 3, we apply the formula: |cv| = |3|·17 = 3·17 = 51. Choice B is correct because it properly applies |cv| = |c|·|v|. Choice C makes an arithmetic error, calculating 17² = 289 instead of 3·17 = 51. When computing from components v = ⟨a, b⟩, remember cv = ⟨ca, cb⟩, and then find magnitude using |cv| = √((ca)² + (cb)²) = |c|√(a² + b²), confirming the formula. Remember: the absolute value in |cv| = |c|·|v| ensures magnitudes are always positive, so even if c = -3, we have |cv| = 3|v|, not -3|v|.

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. When a vector v is multiplied by a scalar c, the magnitude of the result is |cv| = |c|·|v| (the absolute value of c times the magnitude of v), and the direction either stays the same (if c > 0) or reverses 180° (if c < 0). The magnitude is |cv| = |2|·10 = 2·10 = 20, and since c is positive, the direction remains the same as the original vector v's direction of 30°. Choice C is correct because it properly applies |cv| = |c|·|v| and correctly identifies direction based on sign of c. Choice B incorrectly claims the direction reverses to 210° when c = 2, but since c is positive, the direction actually stays the same. Special scalars to remember: c = 1 (no change), c = -1 (flip direction only), c = 2 (double length, same direction), c = -2 (double length, opposite direction), c = 1/2 (half length, same direction). For direction: think of the sign of c as a switch—positive means 'keep the same direction,' negative means 'flip 180° to the opposite direction,' and the magnitude of c only affects how much to scale, not which way to point.

This question tests understanding of how scalar multiplication affects the magnitude and direction of a vector. The formula |cv| = |c|·|v| tells us that the magnitude scales by the absolute value of the scalar: |c| > 1 stretches the vector, 0 < |c| < 1 compresses it, and the absolute value ensures the magnitude is always positive regardless of whether c is positive or negative. Given |v| = 8 and scalar c = -2, we apply the formula: |cv| = |-2|·8 = 2·8 = 16. Choice B is correct because it properly applies |cv| = |c|·|v|. Choice C forgets to take the absolute value of c, computing |cv| = c·|v| = -16, but magnitude must always be positive (|cv| = |c|·|v|). Remember: the absolute value in |cv| = |c|·|v| ensures magnitudes are always positive, so even if c = -3, we have |cv| = 3|v|, not -3|v|. Special scalars to remember: c = 1 (no change), c = -1 (flip direction only), c = 2 (double length, same direction), c = -2 (double length, opposite direction), c = 1/2 (half length, same direction).