The angle between two vectors \( \vec{u} = \langle 3, 4 \rangle \) and \( \vec{v} = \langle -1, 2 \rangle \) is what in degrees?

["Understanding the Angle Between Two Vectors: How to Calculate the Angle Between ( \vec{u} = \langle 3, 4 \rangle ) and ( \vec{v} = \langle -1, 2 \rangle )", "When working with vectors in mathematics and physics, one fundamental concept is determining the angle between them. Whether you're analyzing forces, motion, or directions in space, knowing the angle between two vectors helps clarify their geometric relationship — and this often involves calculating the angle using the dot product formula.", "In this article, we’ll explore the precise angle between two common 2D vectors:\n[\n\vec{u} = \langle 3, 4 \rangle \quad \ ext{and} \quad \vec{v} = \langle -1, 2 \rangle\n]\nand explain the steps to compute it in degrees.", "---", "### Step 1: Formula for the Angle Between Two Vectors", "The angle ( \ heta ) between two vectors ( \vec{u} ) and ( \vec{v} ) is given by:\n[\n\cos \ heta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| \cdot |\vec{v}|}\n]\nThen take the inverse cosine (arccos) to find ( \ heta ):\n[\n\ heta = \arccos\left( \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| \cdot |\vec{v}|} \right)\n]", "---", "### Step 2: Compute the Dot Product ( \vec{u} \cdot \vec{v} )", "The dot product of vectors ( \vec{u} = \langle a_1, a_2 \rangle ) and ( \vec{v} = \langle b_1, b_2 \rangle ) is:\n[\n\vec{u} \cdot \vec{v} = a_1 b_1 + a_2 b_2\n]", "For ( \vec{u} = \langle 3, 4 \rangle ) and ( \vec{v} = \langle -1, 2 \rangle ):\n[\n\vec{u} \cdot \vec{v} = (3)(-1) + (4)(2) = -3 + 8 = 5\n]", "---", "### Step 3: Compute the Magnitudes ( |\vec{u}| ) and ( |\vec{v}| )", "The magnitude of a vector ( \vec{w} = \langle x, y \rangle ) is:\n[\n|\vec{w}| = \sqrt{x^2 + y^2}\n]", "So:\n[\n|\vec{u}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\n]\n[\n|\vec{v}| = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.236\n]", "---", "### Step 4: Plug Into the Cosine Formula", "[\n\cos \ heta = \frac{5}{5 \cdot \sqrt{5}} = \frac{5}{5\sqrt{5}} = \frac{1}{\sqrt{5}} \approx 0.4472\n]", "---", "### Step 5: Compute the Angle in Degrees", "Now take the arccosine and convert to degrees:\n[\n\ heta = \arccos\left( \frac{1}{\sqrt{5}} \right) \approx \arccos(0.4472) \approx 63.43^\circ\n]", "---", "### Final Answer", "The angle between vectors ( \vec{u} = \langle 3, 4 \rangle ) and ( \vec{v} = \langle -1, 2 \rangle ) is approximately 63.43 degrees.", "---", "### Why This Matters", "Understanding vector angles is essential in fields such as physics (for force components), engineering (for mechanical linkages), computer graphics (for lighting and shadows), and data science (for measuring similarity between data points in vector space). Knowing how to compute angles empowers accurate modeling and problem-solving.", "---", "Summary:\n- Use the dot product and vector magnitudes\n- Apply the formula: ( \ heta = \arccos\left( \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} \right) )\n- Calculate step-by-step for clarity\n- Angle between ( \langle 3, 4 \rangle ) and ( \langle -1, 2 \rangle ) ≈ 63.43°", "Mastering this calculation helps unlock deeper insights into vector geometry and application."]