The line ${{l}_{1}}$ passes through the point $A$, whose position vector is $-\mathbf{i}+2\mathbf{j}

Question

The line ${{l}_{1}}$ passes through the point $A$, whose position vector is $-\mathbf{i}+2\mathbf{j}$, and is parallel to the vector $\mathbf{i}+\mathbf{k}$. The line ${{l}_{2}}$ passes through the point $B$, whose position vector is $\mathbf{i}+\mathbf{j}+3\mathbf{k}$, and is parallel to the vector $\mathbf{j}+\mathbf{k}$. Show that the lines ${{l}_{1}}$ and ${{l}_{2}}$ are skew. [2] Find the position vector of the point $N$ on ${{l}_{2}}$ such that $AN$ is perpendicular to ${{l}_{2}}$. [3] The plane $\prod $ contains ${{l}_{2}}$ and is perpendicular to $AN$. Find a vector equation for $\prod $ in the form $\mathbf{r}=\mathbf{u}+a\mathbf{v}+\beta \mathbf{w}$, where $\mathbf{v}$ and $\mathbf{w}$ are perpendicular vectors. [3] The point $X$ varies in such a way that the mid-point of $AX$ is always in $\prod $. Show that $X$ lies on a plane parallel to $\prod $. [4]

Accepted Answer

(i) Lines are skew (shown) (ii) N(0, 2, -1) (iii) π: r·(1,1,-1) = -3 (iv) X lies on plane parallel to π

2016 RI P1 Q11 [Modified]

2016 RI P1 Q11 [Modified]

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