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Question 1 :\[Show\;that\; the \; points\;
with\; P. V.\; \vec{a}-2\vec{b}+3\vec{c},\; \;-2\vec{a}+3\vec{b}+2\vec{c}\;
and\]\[-8\vec{a}+13\vec{b}\; are\; collinear\; vectors.\]Hint: Let given
vectors are the position vectors of point A, B and C. Find vector AB and AC. If one
vector is the scalar multiple of the other then they are called parallel
vectors. Also these are co-initial vectors so these vector are called
collinear vectors. |
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Question 2 : If A is a point (1, -2) and the
vector AB has component 2 and 3, find the coordinates of point
B. \[Hint:\; use \; \; \overrightarrow{AB}=P.V. of B-P.V. of A\; \;
.......\; Ans[3,1] \; \] |
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Question 3 : Using vectors prove that the
points A(-1,2), B(0,0), C(2,-4) are collinear. |
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Question 4 :\[Find \; the \; values\; of\; x, y, z\;
so\; that\; the \; vector\: \vec{a}= x\hat{i}+2\hat{j}+z\hat{k}\]\[and\; \;
\vec{b}=2\hat{i}+y\hat{j}+\hat{k}\; are\; equal \; \; ....\; \;
Ans[x=y=2,z=1]\] |
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Question 5 : \[If\; \; \vec{a}=
\hat{i}+\hat{j}+\hat{k},\: \: \vec{b}= 4\hat{i}-2\hat{j}+3\hat{k},\; \;
\vec{c}= \hat{i}-2\hat{j}+\hat{k},\; find\; a\; vector\; of \;
magnitude\]\[6\; units,\; which\; is\; parallel\; to\; the\; vector\; \;
2\vec{a}-\vec{b}+3\vec{c}.\; \; .......\; Ans[\pm
2(\hat{i}-2\hat{j}+2\hat{k})]\] |
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Question 6 :\[Find\;a \; vector\; \; \vec{a}\; of\;
magnitude\; 5\sqrt{2}\; making\; an\; angle\; of\; \frac{\pi }{4} \; with\; x
- axis,\]\[\frac{\pi }{2} \; with \; y-axis \; and \; an\; acute\; angle\;
\theta\; with\; the \; z-axis.\: \: ...\: Ans[5(\hat{i}+\hat{k})]\] |
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Question 7 :\[Find\;the \; P.V.\;of \;a\; point\;
which\; divides\; the\; join \; of \; points\; with\;
P.V.\]\[\vec{a}-2\vec{b}\; and\; 2\vec{a}+\vec{b}\; externally\; in\; the\;
ratio\; 2:1\: \: ....\: \: Ans[3\vec{a}+4\vec{b}]\] |
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Question 8 :\[If\;two \; vectors\; \vec{a}\;
and \; \vec{b} \; are \; such\; that\; |\vec{a}|=3,\; |\vec{b}|=2\;
and\]\[\vec{a}.\vec{b}=4, \; then\; find \; the\; value\; of\;
(3\vec{a}-4\vec{b}).\; (2\vec{a}+5\vec{b})\; \; .....\; \; Ans[2]\] |
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Question 9 :\[If\;the \; angle\; between \; two \;
vectors \; \vec{a} \;and\; \vec{b} \; of\; equal\;magnitude\; is\; \frac{\pi
}{6} \; and\; their\]\[dot \; product\; is\; 2\sqrt{3}, \; then \; find\;
their\; magnitudes.\; .......\; Ans[|\vec{a}|=2=|\vec{b}|]\] |
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Question 10 :\[Find \; the\; cosine\; of \; the\;
acute\; angle\; which\; the\; vector\; \sqrt{2}\; \hat{i}+\hat{j}+\hat{k}\\
makes\; with\; y-axis.\; \; ......\; \; Ans[1/2]\]Hint: Find the dot
product between the given vector and the unit vector along y-axis. |
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Question 11 :\[Find \; \lambda \; so\; that \; the\;
projection\; of\; \vec{a}=\lambda \hat{i}+\hat{j}+4\hat{k}\;
on\]\[\vec{b}=2\hat{i}+6\hat{j}+3\hat{k}\: \; is\; 4 \; units.\; \; .....\;
\; Ans[5]\] |
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Question 12 :\[Find \; \; |\vec{a}-\vec{b}|, \; if\;
\; |\vec{a}|=2, \; |\vec{b}|=3\; and\; \vec{a}.\vec{b} = 4.\: ....\:
Ans[\sqrt{5}]\] |
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Question 13 :\[If \; |\vec{a}| \; and\; |\vec{b}| \;
are\; two\;unit \; vectors\; and\; |\vec{a}+\vec{b}|\; is \; also\; a\;
unit\; vector\; then\]\[find \; the \; angle\; between\; \vec{a} \; and \;
\vec{b}.\: ....\: Ans\left [\frac{2\pi }{3} \right ]\] |
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Question 14 :\[Let\; \vec{a}, \vec{b}, \vec{c},\;
be\; three\; vectors\; of\; magnitude\; 3,\; 4,\; 5 \; respectively.\; If \;
each \; one \; is\; \perp \]\[ to\; the\; sum \; of \; the\; other \; two\;
vectors, \; then \; prove\; that\; |\vec{a}+\vec{b}+\vec{c}|=5\sqrt{2}\] |
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Question 15 :\[If\; \vec{a}+\vec{b}+\vec{c}=0 \;
and\; |\vec{a}|=5,\; |\vec{b}|=3\; and\; |\vec{c}|=7, \; then \]\[ find\;
the\; angle\; between\; \vec{a} \; and\; \vec{b}\; \; ......\; \; Ans\left [
\frac{\pi }{3} \right ]\]Hint: Bring vector c to the RHS and then
squaring on both side |
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Question 16 :\[Express\; the \; vector\;
\vec{a}=5\hat{i}-2\hat{j}+5\hat{k}\; as\; the\; sum\; of \; two\; vectors\;
such\; that\; one\; is\]\[ parallel\; to \; the\; vector\;
\vec{b}=3\hat{i}+\hat{k}\; and\; the\; other\; is\; \perp\; to \;
\vec{b}\]\[.......\; \; Ans\; [6\hat{i}+2\hat{k}+(-\hat{i}- 2\hat{j}
+3\hat{k})]\]\[Hint:\; Let \; \vec{a}=\vec{c}+\vec{d}, \; find\; \;
\vec{c}=\lambda \; \vec{b}, \; now\; find\; \vec{d} \; by \; using\;
\vec{d}=\vec{a}-\vec{c} \]\[ Now \; find \; the\; value\; of\; \lambda\; by\;
using \; \; \vec{d}.\vec{b}=0\] |
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Question 16 :\[Express\; the \; vector\;
\vec{a}=5\hat{i}-2\hat{j}+5\hat{k}\; as\; the\; sum\; of \; two\; vectors\;
such\; that\; one\; is\]\[ parallel\; to \; the\; vector\;
\vec{b}=3\hat{i}+\hat{k}\; and\; the\; other\; is\; \perp\; to \;
\vec{b}\]\[.......\; \; Ans\; [6\hat{i}+2\hat{k}+(-\hat{i}- 2\hat{j}
+3\hat{k})]\]\[Hint:\; Let \; \vec{a}=\vec{c}+\vec{d}, \; find\; \;
\vec{c}=\lambda \; \vec{b}, \; now\; find\; \vec{d} \; by \; using\;
\vec{d}=\vec{a}-\vec{c} \]\[ Now \; find \; the\; value\; of\; \lambda\; by\;
using \; \; \vec{d}.\vec{b}=0\] |
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Question 17 : \[Express\; the \; vector\;
\vec{a}=6\hat{i}-3\hat{j}-6\hat{k}\; as\; the\; sum\; of \; two\; vectors\;
such\; that\; one\]\[ is \; parallel\; to \; the\; vector\;
\vec{b}=\hat{i}+\hat{i}+\hat{k}\; and\; the\; other\; is\; \perp\; to \;
\vec{b}\]\[.......\: \: Ans\: [-1(\hat{i}+\hat{j} +\hat{k}) +7\hat{i}
-2\hat{j}-5\hat{k}]\] |
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Question 18 :\[Dot \; product\; of \; a\; vector\;
with\; vectors\; 3\hat{i}-5\hat{k}, \; 2\hat{i}+7\hat{j}\; and \;
\hat{i}+\hat{j}+\hat{k} \]\[ are \; respectively\; -1,\; 6 \; and\; 5. \;
Find\; the\; vector....\; \; Ans[3\hat{i}+2\hat{k}]\]Hint : Consider a required
vector in general form with components x, y, z, then find its dot product
with all the given vectors and then solve the different equations |
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Question 19 :\[If\;
\vec{a}=\hat{i}+2\hat{j}+3\hat{k}, \; \vec{b}=-\hat{i}+2\hat{j}+\hat{k}\;
and\; \vec{c}=3\hat{i}+\hat{j}, \; find\; \lambda \]\[ such \; that \;
(\vec{a}+\lambda \vec{b}) \; is \; \perp\; to\; \vec{c}\; \; ......\; \;
Ans[\lambda =5]\]Hint : Two vectors are perpendicular if their dot
product is zero. |
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Question 20 :\[Find\; the \; angle \; between\;
\vec{a} \; and\; \vec{b}\; if\; |\vec{a}|=\sqrt{3},\;\; |\vec{b}|=2\; \;
and\;\; \]\[ \vec{a}.\vec{b}=\sqrt{6}\; \; \; \; \; ........\; \; \; Ans\left
[ \frac{\pi }{4} \right ]\] |
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Question 21 :\[If\;\; \vec{p}=5\hat{i}+\lambda
\hat{j}-3\hat{k}\: and \; \vec{q} =\hat{i} +3 \hat{j} -5\hat{k},\; then\;
find\; the\; value\; of\; \lambda\; so\; that\; \]\[ \vec{p}+\vec{q}\; and\;
\; \vec{p}-\vec{q}\; \; are\; parallel\; vectors.\: .....\: Ans[\lambda =\pm
1]\] |
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Question 22 :\[Let\;\; \vec{a}=\hat{i}+4
\hat{j}+2\hat{k},\: \vec{b} =3\hat{i} -2 \hat{j} +7\hat{k},\;and \;
\vec{c}=2\hat{i}- \hat{j}+4\hat{k}.\; then\; find\; \; \vec{d}\; which\; is
\]\[ perpendicular\; to\;both \; \vec{a} \; and\; \vec{b} \; and\;
\vec{c}.\vec{d}=9.\: .....\: Ans[32\hat{i}-\hat{j}-14\hat{k}]\] |
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Question 23 :\[Let\;\; \vec{a}=4\hat{i}+5
\hat{j}-\hat{k},\: \vec{b} =\hat{i} -4 \hat{j} +5\hat{k},\;and \;
\vec{c}=3\hat{i}+ \hat{j}-\hat{k}.\; then\; find\; \; \vec{d}\; which\; is
\]\[ perpendicular\; to\;both \; \vec{c} \; and\; \vec{b} \; and\;
\vec{d}.\vec{a}=2.\: .....\: Ans[\frac{-2}{63}\left
(\hat{i}-16\hat{j}-13\hat{k} \right )]\] |
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Question 24 :\[Find \; a \; vector\; of\;
magnitude\; 8,\; which\; is \; \perp\; to\; both\; the\; vectors\]\[
2\hat{i}-\hat{j}+3\hat{k}\; and\; -\hat{i}+2\hat{j}-\hat{k}\; \; .......\; \;
Ans\left [ \frac{8}{\sqrt{35}}\left ( -5\hat{i}-\hat{j}+3\hat{k} \right )
\right ]\] |
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Question 25 :\[If\;\vec{a}=
3\hat{i}+2\hat{j}+2\hat{k} \; and \; \vec{b}=\hat{i}+2\hat{j}-2\hat{k},\;
then \; Find \; a \;unit\; vector\; which\; is \; \perp\; \]\[ to\; both\;
the\; vectors\; (\vec{a}-\vec{b}) and (\vec{a}+\vec{b})\; \; .......\; \;
Ans[\frac{1}{3}(-2\hat{i}+2\hat{j}+\hat{k})]\] |
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Question 26 :\[If\;\vec{a}=
2\hat{i}+3\hat{j}-\hat{k}, \; \; \vec{b}=-4\hat{i}+3\hat{j}+2\hat{k},\;and \;
\; \vec{c}=\hat{i}-2\hat{j}-3\hat{k} \; then \]\[ calculate\; (\vec{a}\times
\vec{b}).(\vec{b}\times \vec{c})\; \; ........\; \; Ans [45]\] |
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Question 27 :\[Using \; vectors, \; find \; the\;
area \; of \; the\; triangle \; with \; vertices \]\[ A(1,1,2), B(2,3,5),
C(1,5,5).\: \: \: \: ...........\: \: Ans[\frac{1}{2}\sqrt{61}]\] |
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Question 28 :\[If\;\; \vec{a}\times
\vec{b}=\vec{c}\times \vec{d} \;\; and \; \; \vec{a}\times
\vec{c}=\vec{b}\times \vec{d},\; show\; that\; \vec{a}-\vec{d}\; is\;
parallel\; to\; \; \vec{b}-\vec{c}\]\[Hint:Find \; (\vec{a}-\vec{d})\times
((\vec{b}-\vec{c})),\; then\; using\; the\; given\; conditions,\]\[ if\;
this\; product\; is =0, \; then\; the\; vectors\; are \; \perp \; to \;
each\; other.\] |
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Question 29 :\[Find \; the\; vectors \; of \;
magnitude\; 10\sqrt{3}\; units, \; that\; are\; perpendicular\; to\; the\;
plane\]\[ of \; vectors\; \hat{i}+2\hat{j}+\hat{k}\; and\; -\hat{i}+3\hat{j}+4\hat{k}\;
\; ......\; \; Ans \left (10(\hat{i}-\hat{j}+\hat{k}) \right ).\] |
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Question 30 :\[If\;
\vec{a} \; and\; \vec{b} \; are\; unit\; vectors\; then\; find\; the\;
angle\; between \; \vec{a} \; and \; \vec{b}\]\[ if\; (\sqrt{3}\vec{a}-\vec{b})\;
is \; a \; unit\; vector.\; \; ........\; \; Ans[\frac{\pi }{6}]\] |
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Question 31 :\[If\;\; |\vec{a}\times
\vec{b}|^{2}+(\vec{a}.\vec{b})^{2}=400\; and \; |\vec{a}|=5,\; then\; find\;
|\vec{b}|\; \; ........\; \; Ans[4]\] Hint : Using Lagrange's identity
here. |
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Question 32 :\[If\;\;
\vec{a}=2\hat{i}-3\hat{j}+\hat{k},\; \vec{b}=-\hat{i}+\hat{k}, \;
\vec{c}=2\hat{j}-\hat{k}\; are\; three\; vectors,\; find\; the \; area \]\[
of \; parallelogram \; having\; diagonals\; (\vec{a}+\vec{b}) \; and \;
(\vec{b}+\vec{c})\; \; ......\; \; Ans\left ( \frac{\sqrt{21}}{2} \right )\] |
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Question 33 :\[Two\; adjacent\; sides\; of \; a \;
parallelogram\; are\; \; \hat{i}-2\hat{j}-3\hat{k} \; and \; \;
2\hat{i}-4\hat{j}+5\hat{k}. \]\[ Find \; the \; unit\; vector\; parallel\;
to\; one \; of \; its \; diagonals.\; Also\; find \; its\; area.\]\[\; \;
.....\; \; Ans\left ( \frac{1}{3}(3\hat{i}-6\hat{j} +2\hat{k}), \;
11\sqrt{5}\; sq.unit \right )\] |
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Question 34 : \[Find\; \lambda \; so \; that\;
the\; four \; points\; with \; position\; vectors \; 3\hat{i}+2\hat{j}+
\hat{k},\; 4\hat{i}+\lambda \hat{j}+\; 5\hat{k}\]\[4\hat{i}+2\hat{j}-2
\hat{k}\; and\; 6\hat{i}+5\hat{j}- \hat{k}\; are\; coplanar.\; \; ......\; \;
Ans[\lambda =5]\] |
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Question 35 : \[Find\; the \; volume \; of \;
the\; parallelopiped\; whose \; coterminous\; edges \; are\]\[\vec{a}
=2\hat{i}-3\hat{j}+ 4\hat{k}, \; \vec{b}=\hat{i}+2\hat{j}-1\hat{k} \; and \;
\vec{c}=2\hat{i}-\hat{j}+ 2\hat{k}\: \: .....\: \: Ans[2]\] |
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Question 36 : \[The \; volume \; of \; the\;
parallelopiped\; whose \; coterminous\; edges \; are\;
\vec{a}=-12\hat{i}+\lambda \hat{k}\]\[\vec{b}=3\hat{j}-\hat{k} \; and \;
\vec{c}=2\hat{i}+\hat{j}- 15\hat{k}\; is\; 546\; cubic \; units \; find\;
\lambda \: \: .....\: \: Ans[\lambda =-3]\] |
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Question 37 :Prove the followings\[(i)\; \; [\lambda \vec{a}+\mu \vec{b}\; \; \vec{c}\; \; \vec{d}]=\lambda [\vec{a}\; \; \vec{b}\; \; \vec{c}]+\mu [\vec{b}\; \; \vec{c}\; \; \vec{d}]\]\[(ii)\; \; [\vec{a}+\vec{b}\; \; \vec{b}+\vec{c}\; \; \vec{c}+\vec{a}]= 2[\vec{a}\; \; \vec{b}\; \; \vec{c}]\]\[(iii)\; \; [\vec{a}.(\vec{b}+\vec{c})\times (\vec{a}+2\vec{b}+3\vec{c})]= [\vec{a}\; \; \vec{b}\; \; \vec{c}]\] |
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Question 38 : \[If\; the\; vectors\;
p\hat{i}+\hat{j}+\hat{k},\; \; \hat{i}+q\hat{j}+\hat{k}\; and\;
\hat{i}+\hat{j}+r\hat{k}\; are\; coplanar,\]\[then\; find\; the\; value \;
of\; \; pqr-(p+q+r)\; \; \; ......\; \; Ans[-2]\] |
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Question 39 \[Let\; \vec{a}=\hat{i}-\hat{j},\; \vec{b}=\hat{i}+\hat{j}+\hat{k} \; and\; \vec{c}\; be\; a \; vector\; such\; that\;\; \vec{a}\times \vec{c}+\vec{b}=\vec{0}\]\[and\; \; \vec{a}.\vec{c}=4,\; then \; find \; the \; value \; of\; |\vec{c}|^{2}\; \; .....\; \; Ans\left [\frac{19}{2} \right ]\]\[Hint: Let\; \vec{c}=x\hat{i}+y\hat{j}+z\hat{k}, \; now\; using \; \vec{a\times \vec{c}}=-\vec{b}\; and \; \vec{a}.\vec{c}=4,\; and\; by\]\[compairing\; the\; components\; find \; the \; relations\; between \; x,\; y, \; z.\; \;Solve\; these\]\[relations \; for \; the\; value \; of\; x,\; y,\; z,\; then\; find\; \vec{c} \; then \; its \; magnitude.\] |
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Question 40 \[If \; \; \vec{a}=\hat{i}+2\hat{j}+4\hat{k},\; \;
\vec{b}=\hat{i}+\lambda \hat{j}+4\hat{k} \; \; and \; \;
\vec{c}=2\hat{i}+4\hat{j}+(\lambda ^{2}-1)\hat{k}\]\[be\; copanar\;
vectors.\; Then \; find\; the\; value \; of\; \vec{a}\times \vec{c}\; \; ....\;
\; Ans[-10\hat{i}+5\hat{j}]\] |
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Question 41 \[Let\; \;
\vec{a}=3\hat{i}+2\hat{j}+2\hat{k} \; \; and \; \;
\vec{b}=\hat{i}+2\hat{j}-2\hat{k}\; be\; two\; vectors.\; If\; a \; vector\;
\; \perp\]\[to\; both\; the\; vectors\; \; \vec{a}+\vec{b}\; and\; \vec{a}-\vec{b}\;
has\; the \; magnitude\; 12,\; then\; find\; the\; vector.\]\[\: \: ...\:
Ans[\pm 4(2\hat{i}-2\hat{j}-\hat{k})]\]\[Hint: \; Required \; vector \; is
\;\; \lambda \left [(\vec{a}+\vec{b})\times (\vec{a}-\vec{b}) \right ]\;
and\; \left |\lambda \left [(\vec{a}+\vec{b})\times (\vec{a}-\vec{b}) \right
] \right |=12.\]\[Find \; \lambda \; and\; then\; \; the \; Required \;
vector.\] |
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