a) $u$ and $w$ are parallel
b) $u$ is parallel to $v + w$
c) $v$ and $w$ are orthogonal
d) $u$ is orthogonal to $v + w$
I chose d) since $u\cdot(v+w) = u\cdot v + u\cdot w$.
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Sometimes things depend on what space you are working in! a) Not true. A simple counter-example takes the unit vectors in 3-space $\hat{i},\hat{j},\hat{k}$. All of them dot to zero, but none of them are parallel. b) Not true. Take the same three vectors as above, $\hat{i}+\hat{j}$ lies in the $xy$-plane which is orthogonal to $\hat{k}$ c) Not true. Take $\hat{i}\cdot \hat{k}=0, (\hat{i}+\hat{j})\cdot \hat{k}=0$. d) As you have said, $\cdot $ is associative, so the sum of two vectors orthogonal to $u$ is orthogonal to $u$. |
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You're correct. Specifically, assuming $v \ne w$, $u$ is perpendicular to two independent vectors. These vectors span a plane that contains all vectors of form $<av+bw>$. $u$ is perpendicular to any vector in this plane($u$ is a normal vector to the plane). This can be shown as follows: $$<av+bw>\cdot u=av\cdot u+bw\cdot u=a(0) + b(0) = 0$$ |
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