Q.Find the acute angle between the line $\vec{r}=\lambda(\hat{i}-\hat{j}+\hat{k})$ and the plane $\vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=23$.

The line's direction is perpendicular to the plane's normal (parallel to plane) and a point of the line satisfies the plane's equation — exactly the criterion stated in Reason.

The line $\dfrac{x-1}{3}=\dfrac{y-2}{11}=\dfrac{z+1}{11}$ has direction ratios $(3,11,11)$ and passes through $(1,2,-1)$. The plane $11x-3z-14=0$ has normal $(11,0,-3)$.

Parallel check: $\vec d\cdot\vec n = (3)(11)+(11)(0)+(11)(-3) = 33+0-33=0$, so the line is parallel to the plane.

Point-on-plane check: substitute $(1,2,-1)$: $11(1)-3(-1)-14 = 11+3-14=0$, so the point lies on the plane.

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A line is parallel to a plane when its direction vector is perpendicular to the plane's normal vector, i.e. their dot product is zero.

The line $\vec r=(\hat i-2\hat j+\hat k)+\lambda(2\hat i+\hat j+2\hat k)$ has direction vector $\vec d=(2,1,2)$.

The plane $\vec r\cdot(3\hat i-2\hat j+m\hat k)=14$ has normal vector $\vec n=(3,-2,m)$.

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For the line to lie on the plane, its direction must be perpendicular to the plane's normal and its point must satisfy the plane equation; this gives $k=7$.

The line is $\dfrac{x-4}{1}=\dfrac{y-2}{1}=\dfrac{z-k}{2}$, with direction ratios $(1,1,2)$ and passing through the point $(4,2,k)$.

The plane is $2x-4y+z=7$, with normal vector $(2,-4,1)$.

Condition 1 (line parallel to plane): direction vector $\perp$ normal, i.e. dot product $=0$:

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line parallel to plane ⇒ direction vector ⊥ plane's normal

Direction vector of the line: $(-K,2,1)$. Normal to the plane $\vec r\cdot(2\hat i+3\hat j-4\hat k)+7=0$: $(2,3,-4)$.

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line is parallel to a plane when its direction ratios are perpendicular to the plane's normal

The line has direction ratios $(3,4,5)$. Test each plane's normal $\vec n$ for $\vec n\cdot(3,4,5)=0$:

1. $(2,3,4)$: $6+12+20=38\ne0$
2. $(3,4,5)$: $9+16+25=50\ne0$ …

line parallel to a plane ⇒ direction ratios ⟂ normal

The line $\dfrac{x+R}{3p}=\dfrac{y-3}{2}=\dfrac{z-4}{1}$ has direction ratios $(3p,\,2,\,1)$ — the constant $R$ in the numerator is only a shift of origin along the line and plays no role in its direction, so its (garbled/illegible) value does not affect the answer.

The plane $2x+3y-4z+7=0$ has normal vector $(2,3,-4)$.

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