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arrow_back Determine whether the statements are true or false, and justify your answer

by Platinum
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Determine whether the statements are true or false, and justify your answer

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(a) The vectors $(3,-1,2)$ and $(0,0,0)$ are orthogonal.

 

(b) If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal vectors, then for all nonzero scalars $k$ and $m, k \mathbf{u}$ and $m \mathbf{v}$ are orthogonal vectors.

 

(c) The orthogonal projection of $\mathbf{u}$ on a is perpendicular to the vector component of $\mathbf{u}$ orthogonal to $\mathbf{a}$.

 

(d) If a and b are orthogonal vectors, then for every nonzero vector $\mathbf{u}$, we have $$ \operatorname{proj}_{\mathbf{a}}\left(\operatorname{proj}_{\mathbf{b}}(\mathbf{u})\right)=\mathbf{0} $$

 

(e) If a and $\mathbf{u}$ are nonzero vectors, then $$ \operatorname{proj}_{\mathbf{a}}\left(\operatorname{proj}_{\mathbf{a}}(\mathbf{u})\right)=\operatorname{proj}_{\mathbf{a}}(\mathbf{u}) $$

 

(f) If the relationship $$ \operatorname{proj}_{\mathbf{a}} \mathbf{u}=\operatorname{proj}_{\mathbf{a}} \mathbf{v} $$ holds for some nonzero vector a, then $\mathbf{u}=\mathbf{v}$.

(g) For all vectors $\mathbf{u}$ and $\mathbf{v}$, it is true that $$ \|\mathbf{u}+\mathbf{v}\|=\|\mathbf{u}\|+\|\mathbf{v}\| $$
by Platinum
(106,962 points)

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