Vectors

Concept: In symmetric form: Direction ratios are . Step 1: {\color{red}Identify direction ratios.} Given: Direction ratios: Step 2: {\color{red}Interpret zero component.} Since direction ratio along is zero, line has no movement in -direction. So it lies parallel to plane of constant . Step 3: {\color{red}Geometric interpretation.} Line remains at fixed and varies in x-y plane. Thus it is parallel to z-axis.

Concept: Three vectors are non-coplanar if their scalar triple product is nonzero. Since are non-coplanar, their triple product . We express new vectors in terms of this basis. Step 1: {\color{red}Write vectors in component form.} Let basis be . Step 2: {\color{red}Scalar triple product determinant.} Upper triangular determinant: Step 3: {\color{red}Non-coplanar condition.} So exclude: But since first vector already contains , coplanarity collapses only for one effective value. Thus vectors are non-coplanar for all except one value of .

Concept: Use identity: Expand using dot products: Step 1: {\color{red}Use equal magnitudes.} Let each magnitude = 1 (scaling does not affect cosines). Then: Since square ≥ 0: where . Step 2: {\color{red}Find minimum.} Step 3: {\color{red}Achieving equality.} Minimum occurs when: which is possible for three equal vectors at .

Concept: Given: So is perpendicular to both and . Thus: Step 1: {\color{red}Use magnitude condition.} Since all are unit vectors, find magnitude of : Step 2: {\color{red}Scale to unit vector.} To make magnitude 1: Step 3: {\color{red}Conclusion.}

Concept: Use identities: Step 1: {\color{red}Compute both sides.} LHS: RHS: Step 2: {\color{red}Equate expressions.} Use: Step 3: {\color{red}Simplify.} Test options. For : Closest consistent value ⇒ .

Concept: For an orthogonal matrix: So rows are orthonormal: \begin{itemize} \item Each row has unit length. \item Rows are mutually perpendicular. \end{itemize} Step 1: {\color{red}Row norms = 1.} Row 1: Row 2: Row 3: Step 2: {\color{red}Check orthogonality.} Dot products vanish due to symmetric signs. Closest matching option: