List of top Mathematics Questions on 3D Geometry asked in CUET (UG)

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:

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 \( \vec{a} \) 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:

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:

Find the equation of the plane which passes through the point \[ (1,-2,3), \] contains the line of intersection of the planes \[ x+y+z=1 \] and \[ 2x-y+3z=4, \] and is perpendicular to the plane \[ x-2y+2z+5=0. \]

Line through \((1,2,3)\) parallel to \(\hat{i}+2\hat{j}+3\hat{k}\):

If \(\vec{a}=2\hat{i}+\hat{j}\), \(\vec{b}=\hat{i}+3\hat{j}\), then \(\vec{a}\cdot\vec{b}=\)

Let \[ \vec{a}=2\hat{i}-\hat{j}+\hat{k}, \qquad \vec{b}=\hat{i}+2\hat{j}-\hat{k}, \] and \[ \vec{c}=\lambda\hat{i}+\mu\hat{j}+3\hat{k}. \] If \[ [\vec{a}\ \vec{b}\ \vec{c}]=0 \] and \[ \vec{c}\cdot(\vec{a}+\vec{b})=10, \] then find the value of \(\lambda+\mu\).

The direction cosines of the line which is perpendicular to the lines with direction ratios 1,-2,-2 and 0, 2, 1 are:

The angle between two lines whose direction ratios are proportional to \(1, 1, -2\) and \((\sqrt{3} - 1), (-\sqrt{3} - 1), -4\) is: