Let the pairs and be such that each pair determines a plane. Then the planes are parallel, if

Question

Let the pairs $(\vec{p}, \vec{q})$ and $(\vec{r}, \vec{s})$ be such that each pair determines a plane. Then the planes are parallel, if

Smart Achievers · Accepted Answer

(3)

The planes are parallel, the normal of one plane is perpendicular to any vector of the other plane $\vec{p} \times \vec{q}$ and $(\vec{r}, \vec{s})$ are parallel.

Solution

Q.
Let the pairs $(\vec{p}, \vec{q})$ and $(\vec{r}, \vec{s})$ be such that each pair determines a plane. Then the planes are parallel, if

1 $(\vec{p} \times \vec{q}) \cdot (\vec{r} \times \vec{s}) = 0$

2 $(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{s}) \neq 0$

3 $(\vec{p} \times \vec{q}) \cdot \vec{r} = 0 & (\vec{p} \times \vec{q}) \cdot \vec{s} = 0$

4 $(\vec{p} \times \vec{q}) \cdot \vec{r} \neq 0$

Ans.
(3)

The planes are parallel, the normal of one plane is perpendicular to any vector of the other plane $\vec{p} \times \vec{q}$ and $\vec{r}$ are parallel.

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Solution

Q. Let the pairs (p→,q→) and (r→,s→) be such that each pair determines a plane. Then the planes are parallel, if

1 (p→×q→)·(r→×s→)=0

2 (p→×q→)×(r→×s→)≠0

3 (p→×q→)·r→=0 & (p→×q→)·s→=0

4 (p→×q→)·r→≠0

Ans. (3) The planes are parallel, the normal of one plane is perpendicular to any vector of the other plane p→×q→ and r→ are parallel.