Let u and v be non-collinear vectors in mathb | vedclass

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(B) Since $u$ and $v$ are non-collinear vectors in $\mathbb{R}^2$,they are linearly independent and form a basis for $\mathbb{R}^2$. Thus,any vector i

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Only $(i)$ is true,but (ii) is false

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(B) Since $u$ and $v$ are non-collinear vectors in $\mathbb{R}^2$,they are linearly independent and form a basis for $\mathbb{R}^2$. Thus,any vector in $\mathbb{R}^2$ can be expressed as a linear combination of $u$ and $v$. Statement $(i)$ is true.By definition,the orthogonal projection of $u$ on $v$ is given by $w = \left( \frac{u \cdot v}{|v|^2} ight) v$. This is a scalar multiple of $v$. Since $w$ is a multiple of $v$,it can be written as $w = 0u + \left( \frac{u \cdot v}{|v|^2} ight) v$. For $w$ to be written as $au + bv$ with $a 
eq 0$ and $b 
eq 0$,$w$ would need to have a non-zero component in the direction of $u$. However,$w$ is orthogonal to $u - w$,and $w$ is parallel to $v$. Since $u$ and $v$ are non-collinear,$w$ cannot be expressed as $au + bv$ with $a 
eq 0$. Thus,statement (ii) is false.

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