PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

If the position vector of a point (−4, −3) be \[\vec{a,}\] find \[\left| \vec{a} \right|\]

If the position vector \[\vec{a}\] of a point (12, n) is such that \[\left| \vec{a} \right|\] = 13, find the value (s) of n.

Find the coordinates of the tip of the position vector which is equivalent to \[\vec{A} B\], where the coordinates of A and B are (−1, 3) and (−2, 1) respectively.

If the position vectors of the points A (3, 4), B (5, −6) and C (4, −1) are \[\vec{a,}\] \[\vec{b,}\] \[\vec{c}\] respectively, compute \[\vec{a} + 2 \vec{b} - 3 \vec{c}\].

If \[\vec{a}\] be the position vector whose tip is (5, −3), find the coordinates of a point B such that \[\overrightarrow{AB} =\] \[\vec{a}\], the coordinates of A being (4, −1).

Show that the points 2 \[\hat{i}, -    \hat{i}-4 \] \[\hat{j}\] and \[-\hat{i}+4\hat{j}\]  form an isosceles triangle.

The position vectors of points A, B and C  are \[\lambda \hat{i} +\] 3 \[\hat{j}\],12\[\hat{i} + \mu\] \[\hat{j}\] and 11\[\hat{i} -\] 3 \[\hat{j}\] respectively. If C divides the line segment joining A and B in the ratio 3:1, find the values of \[\lambda\] and \[\mu\]

Find a unit vector in the direction of the resultant of the vectors
\[\hat{i} - \hat{j} + 3 \hat{k} , 2 \hat{i} + \hat{j} - 2 \hat{k} \text{ and }\hat{i} + 2 \hat{j} - 2 \hat{k} .\]

If \[\overrightarrow{PQ} = 3 \hat{i} + 2 \hat{j} - \hat{k}\] and the coordinates of P are (1, −1, 2), find the coordinates of Q.

If the vertices of a triangle are the points with position vectors \[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} , b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} , c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k} ,\]
what are the vectors determined by its sides? Find the length of these vectors.

Find the position vector of a point R which divides the line segment joining points \[P \left( \hat{i} + 2 \hat{j} + \hat{k} \right) \text{ and Q }\left( - \hat{i} + \hat{j} + \hat{k} \right)\] internally 2:1.

Find the position vector of a point R which divides the line segment joining points:

\[P \left( \hat{i} + 2 \hat{j} + \hat{k}\right) \text { and } Q \left( - \hat{i} + \hat{j} + \hat{k} \right)\] externally

Show that the points A, B, C with position vectors \[\vec{a} - 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + 3 \vec{b} - 4 \vec{c}\] and \[- 7 \vec{b} + 10 \vec{c}\] are collinear.

Prove that the points having position vectors \[\hat{i} + 2 \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k} , - 3 \hat{i} - 2 \hat{i} - 5 \hat{k}\] are collinear.

If the points with position vectors \[10 \hat{i} + 3 \hat{j} , 12 \hat{i} - 5 \hat{j}\text{ and a }\hat{i} + 11 \hat{j}\] are collinear, find the value of a.

If \[\vec{a,} \vec{b}\] are two non-collinear vectors prove that the points with position vectors \[\vec{a} + \vec{b,} \vec{a} - \vec{b}\text{ and }\vec{a} + \lambda \vec{b}\] are collinear for all real values of λ.