Q73 / 160 · AP EAMCET 2021 (25 Aug Shift 2) · Mathematics

Let $u$ and $v$ be non-collinear vectors in $R^2$. Let $w$ be the orthogonal projection vector of $u$ on v. Consider two statements : (i) Any vector in $R^2$ can be written as a linear combination of $u$ and $v$ (ii) $w$ can be written as a linear combination of $u$ and $v$ as $w=a u+b v$, where both $a$ and $b$ are non-zero real numbers.

  • A
    Both (i) and (ii) are true
  • Correct answer
    B
    Only (i) is true, but (ii) is false
  • C
    Only (ii) is true, but (i) is false
  • D
    Both (i) and (ii) are false

Solution

Given $\mathbf{u}$ and $\mathbf{v}$ be non-collinear vectors in $R^2$ and $\mathbf{w}$ be the orthogonal projection vector of $\mathbf{u}$ on $\mathbf{v}$. Any vector in $R^2$ can be written as linear combination of $\mathbf{u}$ and $\mathbf{v}$ because $u \neq \lambda v$ for any constant $\lambda$. Hence, $\{\mathbf{u}, \mathbf{v}\}$ set is linearly independent but $\mathbf{w}$ can not be written as $a \mathbf{u}+b \mathbf{v}$ because $\mathbf{w} . \mathbf{u} \neq 0$ and $\mathbf{W} . \mathbf{v} \neq 0$, which is contradiction of orthogonal projection concept.

Hence, the correct answer is option: 'B'.

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