Solution: The cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ is given by $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$. Compute the dot product: $\mathbf{a} \cdot \mathbf{b} = (1)(0) + (0)(1) + (1)(1) = 1$. The magnitudes are $\|\mathbf{a}\| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}$ and $\|\mathbf{b}\| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{2}$. Thus, $\cos\theta = \frac{1}{\sqrt{2} \cdot \sqrt{2}} = \frac{1}{2}$. The final answer is $\boxed{\dfrac{1}{2

["Understanding Vector Angles: Computing the Cosine of the Angle Between Two Vectors", "When analyzing vectors in mathematics and physics, one fundamental question is determining the angle between them. This angle plays a crucial role in fields such as engineering, computer graphics, and data science. The cosine of the angle θ between two vectors a and b is derived from the dot product and vector magnitudes, offering a precise algebraic measure of their alignment.", "### The Mathematical Foundation", "The cosine of the angle θ between vectors a and b is defined by the formula:", "[\n\cos\ heta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}\n]", "Here,\n- The dot product $\mathbf{a} \cdot \mathbf{b}$ computes the sum of the products of corresponding components.\n- The magnitudes $|\mathbf{a}|$ and $|\mathbf{b}|$ are lengths (Euclidean norms) of the vectors.", "### Applying the Formula Step-by-Step", "Let’s apply this formula using specific vectors:\n$$\n\mathbf{a} = \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 0 \ 1 \ 1 \end{pmatrix}\n$$", "Step 1: Compute the dot product\n[\n\mathbf{a} \cdot \mathbf{b} = (1)(0) + (0)(1) + (1)(1) = 0 + 0 + 1 = 1\n]", "Step 2: Compute the magnitude of $\mathbf{a}$\n[\n|\mathbf{a}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{1 + 0 + 1} = \sqrt{2}\n]", "Step 3: Compute the magnitude of $\mathbf{b}$\n[\n|\mathbf{b}| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{0 + 1 + 1} = \sqrt{2}\n]", "Step 4: Plug into the cosine formula\n[\n\cos\ heta = \frac{1}{\sqrt{2} \cdot \sqrt{2}} = \frac{1}{2}\n]", "### Conclusion", "The cosine of the angle between vectors a and b is therefore:", "[\n\boxed{\dfrac{1}{2}}\n]", "This result confirms that the angle θ between a and b is $60^\circ$, since $\cos 60^\circ = \frac{1}{2}$. Understanding this concept allows for deeper analysis of vector relationships across scientific disciplines."]