angle$, find the scalar $k$ such that the vector $\mathbf{p} + k\mathbf{q}$ is perpendicular to $\mathbf{r}$.

["Title: How to Find the Scalar $k$ So That $\mathbf{p} + k\mathbf{q}$ Is Perpendicular to $\mathbf{r}$: A Clear Geometric Approach", "When studying vectors in mathematics and physics, one common problem is finding a scalar $k$ such that a vector combination $\mathbf{p} + k\mathbf{q}$ is perpendicular to a given vector $\mathbf{r}$. This scenario arises in optimization, projections, and equilibrium analysis. Understanding how to determine $k$ rigorously not only strengthens vector algebra skills but also enhances problem-solving across engineering, computer graphics, and physics.", "In this article, we focus on the key condition: two vectors are perpendicular (orthogonal) if their dot product equals zero. We derive a clear formula for $k$ by enforcing orthogonality between $\mathbf{p} + k\mathbf{q}$ and $\mathbf{r}$.", "---", "### The Core Concept: Dot Product and Orthogonality", "Two vectors $\mathbf{a}$ and $\mathbf{b}$ are perpendicular if and only if:\n[\n\mathbf{a} \cdot \mathbf{b} = 0\n]", "Given vectors\n$$\n\mathbf{a} = \mathbf{p} + k\mathbf{q}, \quad \mathbf{b} = \mathbf{r}\n$$\nthey are perpendicular when:\n[\n(\mathbf{p} + k\mathbf{q}) \cdot \mathbf{r} = 0\n]", "---", "### Expand the Dot Product", "Using the distributive property of the dot product:\n[\n(\mathbf{p} + k\mathbf{q}) \cdot \mathbf{r} = \mathbf{p} \cdot \mathbf{r} + k(\mathbf{q} \cdot \mathbf{r}) = 0\n]", "This is a linear equation in $k$:\n[\n\mathbf{p} \cdot \mathbf{r} + k(\mathbf{q} \cdot \mathbf{r}) = 0\n]", "---", "### Solve for $k$", "Rearranging the equation:\n[\nk(\mathbf{q} \cdot \mathbf{r}) = -\mathbf{p} \cdot \mathbf{r}\n]", "Assuming $\mathbf{q} \cdot \mathbf{r} <br/>\ne 0$, we can solve for $k$:\n[\nk = -\frac{\mathbf{p} \cdot \mathbf{r}}{\mathbf{q} \cdot \mathbf{r}}\n]", "This gives the unique scalar $k$ that makes $\mathbf{p} + k\mathbf{q} \perp \mathbf{r}$.", "> Note: If $\mathbf{q} \cdot \mathbf{r} = 0$, then $\mathbf{q}$ is already perpendicular to $\mathbf{r}$, and adding any multiple of $\mathbf{q}$ keeps the vector perpendicular. But in that special case, the solution is not unique — any real $k$ satisfies the condition. However, for most applications, we assume $\mathbf{q} \cdot \mathbf{r} <br/>\ne 0$ so that $k$ is uniquely determined.", "---", "### Geometric Interpretation", "Think of $\mathbf{p} + k\mathbf{q}$ as a vector lying along a line through $\mathbf{p}$ in the direction of $\mathbf{q}$. To “rotate” $\mathbf{p}$ along $\mathbf{q}$ so that the result is perpendicular to $\mathbf{r}$, $k$ balances the components of $\mathbf{p}$ and $\mathbf{q}$ relative to $\mathbf{r}$.", "---", "### Example for Clarity", "Let:\n$$\n\mathbf{p} = \begin{bmatrix} 1 \ 2 \end{bmatrix},\quad \n\mathbf{q} = \begin{bmatrix} 3 \ 1 \end{bmatrix},\quad\n\mathbf{r} = \begin{bmatrix} 0 \ 1 \end{bmatrix}\n$$", "Check:\n$$\n\mathbf{q} \cdot \mathbf{r} = 3 \cdot 0 + 1 \cdot 1 = 1 <br/>\ne 0 \quad \ ext{(valid case)}\n$$\n$$\n\mathbf{p} \cdot \mathbf{r} = 1 \cdot 0 + 2 \cdot 1 = 2\n$$\n$$\nk = -\frac{2}{1} = -2\n$$", "Then $\mathbf{p} + k\mathbf{q} = \begin{bmatrix} 1 \ 2 \end{bmatrix} + (-2)\begin{bmatrix} 3 \ 1 \end{bmatrix} = \begin{bmatrix} -5 \ 0 \end{bmatrix}$", "Dot product with $\mathbf{r}$:\n$$\n\begin{bmatrix} -5 \ 0 \end{bmatrix} \cdot \begin{bmatrix} 0 \ 1 \end{bmatrix} = -5 \cdot 0 + 0 \cdot 1 = 0 \quad \checkmark\n$$", "Confirmed: the vector is perpendicular.", "---", "### Summary", "To find scalar $k$ such that $\mathbf{p} + k\mathbf{q} \perp \mathbf{r}$:\n1. Use the orthogonality condition: $(\mathbf{p} + k\mathbf{q}) \cdot \mathbf{r} = 0$\n2. Expand using linearity: $\mathbf{p} \cdot \mathbf{r} + k(\mathbf{q} \cdot \mathbf{r}) = 0$\n3. Solve for $k$:\n[\nk = -\frac{\mathbf{p} \cdot \mathbf{r}}{\mathbf{q} \cdot \mathbf{r}}, \quad \mathbf{q} \cdot \mathbf{r} <br/>\ne 0\n]", "This elegant solution bridges vector algebra with geometric intuition, empowering students and professionals alike to solve real-world orthogonality problems efficiently.", "---", "Keywords:\n$k$, scalar, vector perpendicularity, dot product, orthogonality, linear algebra, geometry, $\mathbf{p} + k\mathbf{q} \perp \mathbf{r}$, solve for scalar $k$, vector algebra tutorial.", "---", "Also Read:\n- Projection of a vector onto another\n- How to compute dot products geometrically\n- Applications of perpendicular vectors in machine learning and graphics", "---", "By mastering this technique, you gain a powerful tool for analyzing vector relationships—essential in advanced math, physics, engineering, and data science."]