Vector Equation for Straight Line

Q. The value of p so that the vectors (2i-j+k), (i+2j-3k), (3i+pj+5k) are coplanar should be

Q.

The nature of straight lines represented by the equation $4x^2+12xy+9y^2=0,$ is

1. Real and coincident

2. Real and different

3. Imaginary and different

4. None of the above

Q.

The point of the curve $y^2=2(x-3)$ at which the normal is parallel to the line $y-2x+1=0$is

1. $(5,2)$

2. $\left(-\frac{1}{2},-2\right)$

3. $(5,-2)$

4. $\left(\frac{3}{2},2\right)$

Q. If $20$ lines are drawn in a plane such that no two of them are parallel and no three are concurrent, in how many points will they intersect each other?

Q. In a triangle $ABC,$ a point $P$ is chosen on side $\overrightarrow{AB}$ such that $AP:PB=1:4$ and a point $Q$ is chosen on side $\overrightarrow{BC}$ such that $CQ:QB=1:3.$ Line segment $\overrightarrow{CP}$ and $\overrightarrow{AQ}$ intersect at $M.$ If the ratio $\frac{MC}{PC}$ is expressed as a rational number in the lowest term as $\frac{a}{b},$ then $b-a$ equals

Q. Given, two vectors A=-4i+4j+2k and B=2i-j-k. The angle made by (A+B) with i+2j-4k is

Q. Does a line have both direction ratios and direction cosines??

Q. 9. Let vector a = i + j and vector b = 2i - k , then point of intersection of the lines ra =ba and rb = ab is _________? r is arbitrary vector and i, j , k be unit orthonormal vectors.

Q. The equation $z\overline{z}+(2-3i)z+(2+3i)\overline{z}+4=0$ represents a circle of radius

1. $2$ unit
2. $3$ unit
3. $4$ unit
4. $6$ unit

Q.

The equation of the line joining the point $(3,5)$ to the point of intersection of the lines $4x+y-1=0$and $7x-3y-35=0$ is equidistant from the points $(0,0)$ and $(8,34)$.

1. True

2. False

3. Nothing can be said

4. None of these

Q.

The straight line passing through the point of intersection of the straight lines$x-3y+1=0$ and $2x+5y-9=0$ and having infinite Slope and at a distance of $2$ units from the origin, has the equation

1. $x=2$

2. $3x+y–1=0$

3. $y=1$

4. None of these

Q.

The line passing through the point of intersection of $x+y=2,x-y=0$and is parallel to $x+2y=5$is

1. $x+2y=1$

2. $x+2y=2$

3. $x+2y=4$

4. $x+2y=3$

Q.

Let $\alpha$ be the distance between the lines $-x+y=2$ and $x-y=2$and $\beta$ be the distance between the lines $4x-3y=5$ and $6y-8x=1$, then

1. $20\sqrt{2}\beta =11\alpha$

2. $20\sqrt{2}\alpha =11\beta$

3. $11\sqrt{2}\beta =20\alpha$

4. None of the above

Q. The projection of the line joining the points $(3,4,5)$ and $(4,6,3)$ on the line joining the points $(-1,2,4)$ and $(1,0,5)$ is

1. $\frac{4}{3}$
2. $\frac{2}{3}$
3. $\frac{1}{3}$
4. $\frac{1}{2}$

Q. Find the vector equation of the line which passes through the point $(3,4,5)$ and is parallel to the vector $2\hat{i}+2\hat{j}-3\hat{k}.$

Q.

Prove that the relation R defined on set A of all lines as : R={(L1, L2): L1and L2 are parallel lines} is an equivalence relation

Q.

If $x=ay-1=z-2$ and $x=3y-2=bz-2$ lie in the same plane, then the values of $a,b$ are

1. $a=2,b=3$

2. $a=1,b=1$

3. $b=1,a\in R-\left\{0\right\}$

4. $a=3,b=2$

Q.

Show that the three lines with direction cosines

are mutually perpendicular.

Q. The vectors 2i+3j, 5i+6j, . , 8i+xj have their initial points at( 1, 1) ..the value of x so that they terminate on one straight line is???

Q.

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L₁, L₂): L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Q.

Find the vector equation of the line which is parallel to the vector $3\hat{i}-2\hat{j}+6\hat{k}$ and which passes through the point (1, -2, 3).

Q.

Find the equation of the line passing through the points$P(5,1)$ and $Q(1,-1)$.

Q. ntFind The number of values of c such that the straight line 3x+4y=c touches the curve x/2=x+y .n

Q. If $A$ has coordinates $(-1,5)$ and $\overrightarrow{a}$ is a position vector whose tip is $(1,-3)$. Then the coordinates of the point B such that $\overrightarrow{AB}=\overrightarrow{a}$ is

1. $(1,2)$
2. $(0,2)$
3. $(0,-2)$
4. $(2,0)$

Q. The distance of the point $(3,5)$ from the line $2x+3y-14=0$ measured parallel to the line $x-2y=1$ is

1. $\frac{7}{\sqrt{5}}$ units
2. $\frac{7}{\sqrt{13}}$ units
3. $\sqrt{5}$ units
4. $\sqrt{13}$ units

Q. Find the equation of the plane passing through the point (2, 3, 1), given that the direction ratios of the normal to the plane are proportional to 5, 3, 2.

Q.

Vector equation of a straight line passing through a point given by position vector $\overline{a}$ parallel to $\overline{b}$ be is given by $\overline{r}=\overline{a}-\lambda \overline{b},where\overline{r}$ , where $\overline{r}$ is r position vector of a general point on the line and $\lambda ϵIR$

1. True

2. False

Q. 54. Suppose that p , q , r be three non coplanar vectors . Let the components of the vector s along p , q , r be 4 , 3, 5 respectively . If the components of s along (-p+q+r ) , (p-q+r ) , (-p-q+r ) are x , y , z respectively , then find 2x+y+z .

Q.

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector $2\hat{i}-\hat{j}+4\hat{k}$ and is in the direction $\hat{i}+2\hat{j}-\hat{k}$.

Q. The cartesian from of equation a line passing through the point position vector $2\hat{i}-\hat{j}+2\hat{k}$ and is in the direction of $-2\hat{i}+\hat{j}+\hat{k}$, is

1. $\frac{x-2}{-2}=\frac{y+1}{1}=\frac{z-2}{1}$
2. $\frac{x+4}{-2}=\frac{y-1}{1}=\frac{z+2}{1}$
3. $\frac{x+2}{4}=\frac{y-1}{-1}=\frac{z-1}{2}$
4. $Noneofthese$