This problem requires scalar multiplication of a vector. When multiplying vector $\langle a, b \rangle$ by scalar k, the result is $\langle ka, kb \rangle$. For 4b = 4$\langle 0, 7 \rangle$, we multiply each component by 4: $\langle 4 \cdot 0, 4 \cdot 7 \rangle$ = $\langle 0, 28 \rangle$. Scalar multiplication affects each component independently.

This problem asks for the magnitude of vector v = ⟨5, 12⟩. The magnitude of angle brackets a comma b equals square root of (a squared plus b squared). Calculating: magnitude of angle brackets 5 comma 12 equals square root of (5 squared plus 12 squared) equals square root of (25 plus 144) equals square root of 169 equals 13. Choice C gave 25 (forgot to add 144 before taking square root).

We need to find the magnitude of vector ⟨3, 4⟩. The magnitude formula is: magnitude of ⟨a, b⟩ equals square root of (a squared plus b squared). Calculating: magnitude equals square root of (3 squared plus 4 squared) equals square root of (9 plus 16) equals square root of 25 equals 5. Choice D shows 25, which is the value before taking the square root.

We need to add vectors v = ⟨-2, 5⟩ and w = ⟨6, -1⟩. Vector addition formula: ⟨a, b⟩ plus ⟨c, d⟩ equals ⟨a plus c, b plus d⟩. Calculating: v plus w equals ⟨-2, 5⟩ plus ⟨6, -1⟩ equals ⟨-2 plus 6, 5 plus (-1)⟩ equals ⟨4, 4⟩. Add corresponding components separately.

This problem asks for vector subtraction v - w where v = ⟨7, 1⟩ and w = ⟨1, 7⟩. For vector subtraction, angle brackets a comma b minus angle brackets c comma d equals angle brackets a minus c comma b minus d. Calculating: v minus w equals angle brackets 7 comma 1 minus angle brackets 1 comma 7 equals angle brackets 7 minus 1 comma 1 minus 7 equals angle brackets 6 comma negative 6. Subtract corresponding components.

This problem involves vector addition. When adding vectors ⟨a, b⟩ + ⟨c, d⟩, the result equals ⟨a + c, b + d⟩. For v + w = ⟨1, 2⟩ + ⟨3, 4⟩, we add corresponding components: ⟨1 + 3, 2 + 4⟩ = ⟨4, 6⟩. Vector addition requires adding components separately.

This problem asks for the magnitude of a vector. The magnitude of vector $\langle a, b \rangle$ equals $\sqrt{a^2 + b^2}$. For c = $\langle 8, 15 \rangle$, the magnitude equals $\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$. This is a Pythagorean triple $(8, 15, 17)$.

We need to find the magnitude of vector ⟨8, 6⟩. Using the magnitude formula: magnitude of ⟨a, b⟩ equals square root of (a squared plus b squared). Calculating: magnitude equals square root of (8 squared plus 6 squared) equals square root of (64 plus 36) equals square root of 100 equals 10. This is a 6-8-10 Pythagorean triple (scaled version of 3-4-5).

We need to find -3v where v = ⟨2, -4⟩. Scalar multiplication formula: k times ⟨a, b⟩ equals ⟨ka, kb⟩. Calculating: -3 times ⟨2, -4⟩ equals ⟨-3 times 2, -3 times (-4)⟩ equals ⟨-6, 12⟩. Note that -3 times (-4) gives positive 12.

This problem asks for scalar multiplication 4v where v = ⟨0, 5⟩. For scalar multiplication, k times angle brackets a comma b equals angle brackets ka comma kb. Calculating: 4 times angle brackets 0 comma 5 equals angle brackets 4 times 0 comma 4 times 5 equals angle brackets 0 comma 20. Multiply each component by the scalar 4.