List of top Mathematics Questions on 3D Geometry

If the angle \(\theta\) between the line \[ \frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2} \] and the plane \[ 2x-y+\sqrt{\lambda}z+4=0 \] is such that \(\sin\theta=\frac13\), then the value of \(\lambda\) is:

If \((K,3,5)\), \((2,-1,2)\) are direction ratios of two lines and the angle between them is \(45^\circ\), then a value of \(K\) is:

If \(A(1,1,1)\), \(B(2,3,4)\) and \(C(2,5,7)\) are the vertices of \(\triangle ABC\), then the length of the altitude drawn through the vertex \(A\) is:

If the image of point \((1,-1,1)\) in the plane \[ x-2y+3z=4 \] is \((x_1,y_1,z_1)\), then \(x_1-y_1-z_1=\)

Let D be harmonic conjugate of point C with respect to points \[ A(1,-3,5),\qquad B(5,-3,1) \] If C divides AB in ratio \(3:5\), then point dividing CD in ratio \(1:2\) is

If \((1,-2,2)\) and \((2,6,-3)\) are the direction ratios of two straight lines, then the direction cosines of the line bisecting an angle between these two lines are

A line passes through (2, 1, 3) and (1, 2, -1), then
(A) Equation is \( \frac{x-2}{1} = \frac{y-1}{1} = \frac{z-3}{4} \)
(B) Equation is \( \frac{x+2}{-1} = \frac{y+1}{1} = \frac{z+3}{4} \)
(C) Equation is \( \vec{r} = 2\vec{i} + \vec{j} + 3\vec{k} + \lambda(\vec{i} - \vec{j} + 4\vec{k}) \)
(D) Equation is \( \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z+1}{4} \)
Choose the correct answer:

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}| = 2 \), \( |\vec{b}| = 1 \) and \( \vec{a} \cdot \vec{b} = \sqrt{3} \) then the angle between \( 2\vec{b} \) and \( -\vec{a} \) is:

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a}+\vec{b}$ is also a unit vector. Then which of the following are TRUE?} (A) $|\vec{a}-\vec{b}|=0$
(B) $|\vec{a}-\vec{b}|=\sqrt{3}$
(C) Angle between $\vec{a}$ and $\vec{b}=\frac{2\pi}{3}$
(D) Angle between $\vec{a}$ and $\vec{b}=\frac{\pi}{3}$

If $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2\hat{j}+\hat{k}$ then Match List-I with List-II:}

If a unit vector makes equal acute angles with the coordinate axes, then the projection of this vector on $-5\mathbf{i}+7\mathbf{j}-\mathbf{k}$ is:

If N is the foot of the perpendicular drawn from the point \(P(5,-1,3)\) to the line passing through the points \(A(1,3,-5)\) and \(B(3,-1,5)\) then the ratio in which N divides AB is

If l, m, n are the direction cosines of a normal drawn to the plane \(2x-3y+6z-7=0\) and d is the length of the perpendicular drawn from origin to this plane then \(7d|l+m+n|=\)

\(A(-4,9,k)\), \(B(-1,k,k)\), \(C(0,7,10)\) form an isosceles right-angled triangle. If \(AB=BC\) and \(AC\) is an integer then the perimeter of \(\Delta ABC\) is

The square of the distance of the point P(5, 6, 7) from the line $\frac{x-2}{2} = \frac{y-5}{3} = \frac{z-2}{4}$ is equal to:

Let a line \(L\) passing through the point \((1,1,1)\) be perpendicular to both the vectors \(2\hat{i}+2\hat{j}+\hat{k}\) and \(\hat{i}+2\hat{j}+2\hat{k}\). If \((a,b,c)\) is the foot of perpendicular from the origin on the line \(L\), then the value of \(34(a+b+c)\) is:

If the distance of the point $(a, 2, 5)$ from the image of the point $(1, 2, 7)$ in the line $\frac{x}{1} = \frac{y-1}{1} = \frac{z-2}{2}$ is 4, then the sum of all possible values of $a$ is equal to :

Let a triangle PQR be such that P and Q lie on the line $\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$ and are at a distance of 6 units from R(1, 2, 3). If $(\alpha, \beta, \gamma)$ is the centroid of $\Delta PQR$, then $\alpha + \beta + \gamma$ is equal to :

Let the image of the point \(P(0,-5,0)\) in the line \[ \frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2} \] be the point \(R\) and the image of the point \(Q(0,-\frac12,0)\) in the line \[ \frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1} \] be the point \(S\). Then the square of the area of the parallelogram \(PQRS\) is ______.

If the point of intersection of the lines \[ \frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7} \] \[ \frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7} \] lies on the \(xy\)-plane, then the value of \(a+b\) is:

Let \(P\) be the plane such that it contains the straight line \[ \frac{x-1}{2}=\frac{y-3}{3}=\frac{z+2}{1} \] and is perpendicular to the plane \[ x+2y+3z=4 \] Let \(P_1\) be the plane which passes through the point \((4,2,2)\) and is parallel to \(P\). Then which of the following statements is (are) TRUE?

If the point P which divides the line segment joining $A(1,1,1)$ and $B(2,2,2)$ in the ratio 1: m lies on the plane $x+2y+3z-1=0$, then $m=$

If $\theta$ is the acute angle between a line $\frac{x-1}{1}=\frac{y-1}{-1}=\frac{z-1}{1}$ and a normal to the plane $2x+3y+4z=0$ then $\tan^{2}\theta+sec^{2}\theta=$

The points (0, $\lambda$, 1), ($\mu$, 3, -1), ($\lambda$, 5, 0), ($\mu$, 6, $\mu$) taken in that order, form a square. If $\lambda$, $\mu$ are positive real numbers, then the length of its side is

The acute angle between the planes $2x - y + z = 6$ and $x + y + 2z = 3$ is: