Direction Vector of Line L
Line L is perpendicular to two given lines. Their direction vectors are $\vec{d_1} = <4, 1, 1>$ and $\vec{d_2} = <1, 1, 0>$. The direction vector $\vec{d_L}$ of line L is found using the cross product of $\vec{d_1}$ and $\vec{d_2}$.
$\vec{d_L} = \vec{d_1} \times \vec{d_2} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 1 & 1 \\ 1 & 1 & 0 \end{vmatrix}$ $= \mathbf{i}(1 \cdot 0 - 1 \cdot 1) - \mathbf{j}(4 \cdot 0 - 1 \cdot 1) + \mathbf{k}(4 \cdot 1 - 1 \cdot 1)$ $= \mathbf{i}(-1) - \mathbf{j}(-1) + \mathbf{k}(3) = <-1, 1, 3>$
Line L Equation and Point Q
Line L passes through $P(1, 1, 1)$ and has direction vector $\vec{d_L} = <-1, 1, 3>$. The parametric equation for L is $\vec{r} = <1, 1, 1> + t<-1, 1, 3>$, or: $x = 1 - t$ $y = 1 + t$ $z = 1 + 3t$
Point Q is where line L intersects the yz-plane ($x=0$). Set $x=0$: $1 - t = 0 \implies t = 1$. Calculate Q coordinates using $t=1$: $Q = (1-1, 1+1, 1+3(1)) = (0, 2, 4)$.
Line L' Equation and Point R
Line L' is parallel to L, so its direction vector is also $\vec{d_L} = <-1, 1, 3>$. Line L' passes through $S(1, 0, -1)$. The parametric equation for L' is $\vec{r} = <1, 0, -1> + u<-1, 1, 3>$, or: $x = 1 - u$ $y = u$ $z = -1 + 3u$
Point R is where line L' intersects the yz-plane ($x=0$). Set $x=0$: $1 - u = 0 \implies u = 1$. Calculate R coordinates using $u=1$: $R = (1-1, 1, -1+3(1)) = (0, 1, 2)$.
Area of Parallelogram PQRS
The vertices of the parallelogram are $P(1, 1, 1)$, $Q(0, 2, 4)$, $R(0, 1, 2)$, $S(1, 0, -1)$. We use the vectors representing two adjacent sides originating from P: $\vec{PQ}$ and $\vec{PS}$.
- $\vec{PQ} = Q - P = <0-1, 2-1, 4-1> = <-1, 1, 3>$
- $\vec{PS} = S - P = <1-1, 0-1, -1-1> = <0, -1, -2>$
The area of the parallelogram is the magnitude of the cross product $||\vec{PQ} \times \vec{PS}||$.
$\vec{PQ} \times \vec{PS} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 1 & 3 \\ 0 & -1 & -2 \end{vmatrix}$ $= \mathbf{i}(1(-2) - 3(-1)) - \mathbf{j}((-1)(-2) - 3(0)) + \mathbf{k}((-1)(-1) - 1(0))$ $= \mathbf{i}(-2 + 3) - \mathbf{j}(2 - 0) + \mathbf{k}(1 - 0)$ $= <1, -2, 1>$
The area is $||<1, -2, 1>|| = \sqrt{1^2 + (-2)^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6}$.
The square of the area is $(\sqrt{6})^2 = 6$.
Similar Questions
- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to _______.
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
- Consider the parabola $\text{P} : y^2 = 4kx$ and the ellipse $\text{E} : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.
- Let the centre of the circle $x^2 + y^2 + 2gx + 2fy + 25 = 0$ be in the first quadrant and lie on the line $2x - y = 4$. Let the area of an equilateral triangle inscribed in the circle be $27\sqrt{3}$. Then the square of the length of the chord of the circle on the line $x = 1$ is_________.
- If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$, then $\vec{c} \cdot (\vec{a} - 2\vec{b})$ is equal to _________.
- For some $\theta \in \left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2 - y^2\sec^2\theta = 8$ be $e_1$ and $l_1$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2\sec^2\theta + y^2 = 6$ be $e_2$ and $l_2$, respectively. If $e_1^2 = e_2^2(\sec^2\theta+1)$, then $\left(\frac{l_1 l_2}{e_1 e_2}\right)\tan^2\theta$ is equal to ________
- If foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{12} = 1$ and the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ coincide, then the eccentricity of the hyperbola, with foci $(0, \pm \sqrt{10})$ passing through the point $(a, a + 1)$, is
- Let the curves $y^2 = kx$ and $xy = -1$ have a common tangent, whose slope is $\frac{1}{2}$. Then $k$ can NOT lie in the interval
Important Questions from Coordinate Geometry
- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to _______.
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
- Consider the parabola $\text{P} : y^2 = 4kx$ and the ellipse $\text{E} : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.
