Chapter " Vectors" PAST PAPER QUESTIONS XI Mathematics AkUEB Examinations 2026

Vector Concepts and Unit Vectors
📌 A unit vector has a magnitude of 1. To find the unknown variable in a vector like $i - aj$, calculate the magnitude using $\sqrt{1² + (-a)²} = 1$, which simplifies to $a = 0$.
📌 For a vector with a single term like $5/p k$, equate the magnitude to 1 to find that $p = 5$.
📌 Orthogonal vectors have a dot product equal to 0. Use the dot product calculation ($i \cdot i + j \cdot j + k \cdot k = 0$) to solve for unknown variables like $b$.

Parallel Vectors and Scalar Triple Products
📌 Two vectors are parallel if their components are proportional or if their cross product is 0. For vectors $2i - j$ and $xi - 3j$, setting the ratios equal ($\frac{2}{x} = \frac{-1}{-3}$) yields $x = 6$.
📌 The scalar triple product involves calculating the determinant of a 3x3 matrix. This process is essential for determining volume-related properties in vector spaces.

Projections and Dot Products
📌 To calculate the projection of vector A on B, use the formula $\frac{A \cdot B}{|B|}$. For example, with $A = 2i - j$ and $B = i - j$, the projection is calculated as $\frac{3}{\sqrt{2}}$.
📌 Distinguish between scalar projection (which yields a numerical value) and vector projection (which yields a vector result using the formula $\frac{A \cdot B}{|B|²} \cdot B$).
📌 The dot product is calculated by multiplying corresponding components: $i$ with $i$, $j$ with $j$, and $k$ with $k$, then summing the results.

Directional Cosines
📌 Directional cosines are found by dividing each component of a vector by its magnitude: $\cos(\alpha) = \frac{x}{|r|}$, $\cos(\beta) = \frac{y}{|r|}$, and $\cos(\gamma) = \frac{z}{|r|}$.
📌 Ensure all theoretical concepts are mastered before attempting past papers to avoid confusion during complex calculations.

Key Points & Insights
➡️ Master the theory first: Concepts like cross products, dot products, and magnitudes are the foundation; solving past papers without conceptual clarity will lead to errors.
➡️ Check the context: Determine if the problem asks for a scalar or vector projection, as the formulas differ; scalar projection provides a magnitude, while vector projection provides a directional component.
➡️ Verification: When solving for unknowns where magnitudes are equal, equate the magnitude formulas ($\sqrt{x_1² + y_1² + z_1²} = \sqrt{x_2² + y_2² + z_2²}$) and solve for the variable by squaring both sides to remove the root.

📸 Video summarized with SummaryTube.com on Mar 28, 2026, 15:56 UTC