} = 5 $. When multiplied by scalar $ k = -2 $, the new magnitude is $ |kv| = |-2| \cdot 5 = 10 $. Choice B is correct because it correctly calculates magnitude as $ |k| $ times original magnitude. Choice A forgets to take the absolute value of $ k $ when calculating magnitude, using $ |kv| = k |v| $ instead of $ |k| \cdot |v| $, which gives a negative magnitude when $ k $ is negative. Key to scalar multiplication: multiply every component by the scalar ($ kv = \langle ka, kb \rangle $), and remember that the magnitude scales by $ |k| $ (the absolute value) while direction stays the same if $ k > 0 $ or reverses if $ k < 0 $. Special scalars to remember: $ k = 1 $ leaves the vector unchanged, $ k = -1 $ reverses direction only (same length), $ k = 2 $ doubles the length, $ k = 1/2 $ halves the length, and $ k = 0 $ gives the zero vector.29:T474,This question tests understanding of how scalar multiplication affects a vector visually and algebraically. The visual effect of multiplying a vector by scalar k depends on k's value: k > 1 stretches the vector (longer arrow), 0 < k < 1 compresses it (shorter arrow), k < 0 reverses the direction and scales by |k|, and k = 0 gives the zero vector. Multiplying by k = 1/3 which is between 0 and 1 causes the vector to compress by a factor of 1/3, so the arrow becomes shorter compared to the original. Choice A is correct because it accurately describes both length and direction changes. Choice B claims the direction reverses when k is positive, but direction only reverses when k < 0—positive scalars preserve direction. Key to scalar multiplication: multiply every component by the scalar (kv = ⟨ka, kb⟩), and remember that the magnitude scales by |k| (the absolute value) while direction stays the same if k > 0 or reverses if k < 0. Special scalars to remember: k = 1 leaves the vector unchanged, k = -1 reverses direction only (same length), k = 2 doubles the length, k = 1/2 halves the length, and k = 0 gives the zero vector.2a:T48b,This question tests understanding of how scalar multiplication affects a vector visually and algebraically. Scalar multiplication affects vectors component-wise: if v = ⟨a, b⟩, then kv = ⟨ka, kb⟩, meaning each component is multiplied by the scalar k, which visually corresponds to stretching or compressing the arrow by factor |k| and possibly reversing its direction if k is negative. Given v = ⟨6, 2⟩ and scalar k = 1/2, we compute kv = ⟨(1/2)·6, (1/2)·2⟩ = ⟨3, 1⟩, which visually means the endpoint moves from (6, 2) to (3, 1). Choice B is correct because it properly multiplies each component by k. Choice A uses the reciprocal 1/k instead of k, which would scale by the wrong factor. Key to scalar multiplication: multiply every component by the scalar (kv = ⟨ka, kb⟩), and remember that the magnitude scales by |k| (the absolute value) while direction stays the same if k > 0 or reverses if k < 0. Visually, think of scalar multiplication as stretching (if |k| > 1) or compressing (if |k| < 1) the arrow representing the vector, and flipping it 180° if k is negative—the arrow always remains on the same line through the origin.2b:T477,This question tests understanding of how scalar multiplication affects a vector visually and algebraically. Scalar multiplication preserves collinearity: the vector kv is always parallel (if k > 0) or anti-parallel (if k < 0) to the original vector v, meaning they lie on the same line through the origin. Since w is collinear with v but points in the opposite direction and is three times as long, k must be negative with |k| = 3, so k = -3. Choice B is correct because it accurately describes both length and direction changes. Choice A claims the direction remains the same when k is positive, but direction only reverses when k < 0—positive scalars preserve direction. To find the scalar k given two vectors where w = kv, divide any component of w by the corresponding component of v (e.g., if v = ⟨3, 4⟩ and w = ⟨6, 8⟩, then k = 6/3 = 2 or k = 8/4 = 2), or compare magnitudes using k = |w|/|v|. Visually, t

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Given the vector $\mathbf{v}=\langle 3,4\rangle$ and scalar multiplication by $k=2$, what are the components of $2\mathbf{v}$?

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