### Common Errors in Secondary Mathematics

Common Errors Committed  by the  Students  in Secondary Mathematics   Errors  that students often make in doing secondary mathematics  during their practice and during the examinations  and their remedial measures are well explained here stp by step.  Some Common Errors in Mathematics

### Math Assignment | Class XII | Ch-10 | Vector Algebra

Maths Assignment Class 12
Vector Algebra Chapter 10
Extra questions important for the examination. Maths assignment  on vector algebra for complete knowledge of the concept.

Important steps that students need to do
• First of all students should learn and  revise all basic concepts and formulas of vector algebra.
• Revise Chapter 10 vector algebra  NCERT book of Mathematics.
• Revise all examples of chapter 10 of  NCERT book.
• Now start the Assignment.
Assignment on Vector Algebra
Question 1 :
Show that the points with position vector  $\vec{a}-2\vec{b}+3\vec{c},\;\;-2\vec{a}+3\vec{b}+2\vec{c}$  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.

Question 2 : If A is a point (1, -2) and the vector AB has component 2 and 3, find the coordinates of point B.

Ans: (3, 1)

Hint : $\overrightarrow{AB}=P.V.of B-P.V.of A$

Question 3 : Using vectors prove that the points A(-1, 2), B(0, 0), C(2, -4) are collinear.

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, x = 1

 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$$6\; units,\; which\; is\; parallel\; to\; the\; vector\; \; 2\vec{a}-\vec{b}+3\vec{c}.\; \; .......\; Ans[\pm 2(\hat{i}-2\hat{j}+2\hat{k})]$ Question 6 :$Find\;a \; vector\; \; \vec{a}\; of\; magnitude\; 5\sqrt{2}\; making\; an\; angle\; of\; \frac{\pi }{4} \; with\; x - axis,$$\frac{\pi }{2} \; with \; y-axis \; and \; an\; acute\; angle\; \theta\; with\; the \; z-axis.\: \: ...\: Ans[5(\hat{i}+\hat{k})]$ 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}]$

Questions based of the scalar or dot product
 Question 8 :$If\;two \; vectors\; \vec{a}\; and \; \vec{b} \; are \; such\; that\; |\vec{a}|=3,\; |\vec{b}|=2\; and$$\vec{a}.\vec{b}=4, \; then\; find \; the\; value\; of\; (3\vec{a}-4\vec{b}).\; (2\vec{a}+5\vec{b})\; \; .....\; \; Ans[2]$ Question 9 :$If\;the \; angle\; between \; two \; vectors \; \vec{a} \;and\; \vec{b} \; of\; equal\;magnitude\; is\; \frac{\pi }{6} \; and\; their$$dot \; product\; is\; 2\sqrt{3}, \; then \; find\; their\; magnitudes.\; .......\; Ans[|\vec{a}|=2=|\vec{b}|]$ 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. Question 11 :$Find \; \lambda \; so\; that \; the\; projection\; of\; \vec{a}=\lambda \hat{i}+\hat{j}+4\hat{k}\; on$$\vec{b}=2\hat{i}+6\hat{j}+3\hat{k}\: \; is\; 4 \; units.\; \; .....\; \; Ans[5]$ Question 12 :$Find \; \; |\vec{a}-\vec{b}|, \; if\; \; |\vec{a}|=2, \; |\vec{b}|=3\; and\; \vec{a}.\vec{b} = 4.\: ....\: Ans[\sqrt{5}]$ Question 13 :$If \; |\vec{a}| \; and\; |\vec{b}| \; are\; two\;unit \; vectors\; and\; |\vec{a}+\vec{b}|\; is \; also\; a\; unit\; vector\; then$$find \; the \; angle\; between\; \vec{a} \; and \; \vec{b}.\: ....\: Ans\left [\frac{2\pi }{3} \right ]$ Question 14 :$Let\; \vec{a}, \vec{b}, \vec{c},\; be\; three\; vectors\; of\; magnitude\; 3,\; 4,\; 5 \; respectively.\; If \; each \; one \; is\; \perp$$to\; the\; sum \; of \; the\; other \; two\; vectors, \; then \; prove\; that\; |\vec{a}+\vec{b}+\vec{c}|=5\sqrt{2}$ Question 15 :$If\; \vec{a}+\vec{b}+\vec{c}=0 \; and\; |\vec{a}|=5,\; |\vec{b}|=3\; and\; |\vec{c}|=7, \; then$$find\; the\; angle\; between\; \vec{a} \; and\; \vec{b}\; \; ......\; \; Ans\left [ \frac{\pi }{3} \right ]$Hint: Bring vector c to the RHS and then squaring on both side Question 16 :$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$ 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$ 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}]$ 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}$$are \; respectively\; -1,\; 6 \; and\; 5. \; Find\; the\; vector....\; \; Ans[3\hat{i}+2\hat{k}]$Hint : Consider a required vector in general form with components x, y, z, then find its dot product with all the given vectors and then solve the different equations Question 19 :$If\; \vec{a}=\hat{i}+2\hat{j}+3\hat{k}, \; \vec{b}=-\hat{i}+2\hat{j}+\hat{k}\; and\; \vec{c}=3\hat{i}+\hat{j}, \; find\; \lambda$$such \; that \; (\vec{a}+\lambda \vec{b}) \; is \; \perp\; to\; \vec{c}\; \; ......\; \; Ans[\lambda =5]$Hint : Two vectors are perpendicular if their dot product is zero. Question 20 :$Find\; the \; angle \; between\; \vec{a} \; and\; \vec{b}\; if\; |\vec{a}|=\sqrt{3},\;\; |\vec{b}|=2\; \; and\;\;$$\vec{a}.\vec{b}=\sqrt{6}\; \; \; \; \; ........\; \; \; Ans\left [ \frac{\pi }{4} \right ]$
Questions based on the Vector or cross  product
 Question 21 :$If\;\; \vec{p}=5\hat{i}+\lambda \hat{j}-3\hat{k}\: and \; \vec{q} =\hat{i} +3 \hat{j} -5\hat{k},\; then\; find\; the\; value\; of\; \lambda\; so\; that\;$$\vec{p}+\vec{q}\; and\; \; \vec{p}-\vec{q}\; \; are\; parallel\; vectors.\: .....\: Ans[\lambda =\pm 1]$ 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$$perpendicular\; to\;both \; \vec{a} \; and\; \vec{b} \; and\; \vec{c}.\vec{d}=9.\: .....\: Ans[32\hat{i}-\hat{j}-14\hat{k}]$ Question 23 :$Let\;\; \vec{a}=4\hat{i}+5 \hat{j}-\hat{k},\: \vec{b} =\hat{i} -4 \hat{j} +5\hat{k},\;and \; \vec{c}=3\hat{i}+ \hat{j}-\hat{k}.\; then\; find\; \; \vec{d}\; which\; is$$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 )]$ Question 24 :$Find \; a \; vector\; of\; magnitude\; 8,\; which\; is \; \perp\; to\; both\; the\; vectors$$2\hat{i}-\hat{j}+3\hat{k}\; and\; -\hat{i}+2\hat{j}-\hat{k}\; \; .......\; \; Ans\left [ \frac{8}{\sqrt{35}}\left ( -5\hat{i}-\hat{j}+3\hat{k} \right ) \right ]$ Question 25 :$If\;\vec{a}= 3\hat{i}+2\hat{j}+2\hat{k} \; and \; \vec{b}=\hat{i}+2\hat{j}-2\hat{k},\; then \; Find \; a \;unit\; vector\; which\; is \; \perp\;$$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})]$ 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$$calculate\; (\vec{a}\times \vec{b}).(\vec{b}\times \vec{c})\; \; ........\; \; Ans [45]$ Question 27 :$Using \; vectors, \; find \; the\; area \; of \; the\; triangle \; with \; vertices$$A(1,1,2), B(2,3,5), C(1,5,5).\: \: \: \: ...........\: \: Ans[\frac{1}{2}\sqrt{61}]$ 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.$ Question 29 :$Find \; the\; vectors \; of \; magnitude\; 10\sqrt{3}\; units, \; that\; are\; perpendicular\; to\; the\; plane$$of \; vectors\; \hat{i}+2\hat{j}+\hat{k}\; and\; -\hat{i}+3\hat{j}+4\hat{k}\; \; ......\; \; Ans \left (10(\hat{i}-\hat{j}+\hat{k}) \right ).$ Question 30 :$If\; \vec{a} \; and\; \vec{b} \; are\; unit\; vectors\; then\; find\; the\; angle\; between \; \vec{a} \; and \; \vec{b}$$if\; (\sqrt{3}\vec{a}-\vec{b})\; is \; a \; unit\; vector.\; \; ........\; \; Ans[\frac{\pi }{6}]$ 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. 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$$of \; parallelogram \; having\; diagonals\; (\vec{a}+\vec{b}) \; and \; (\vec{b}+\vec{c})\; \; ......\; \; Ans\left ( \frac{\sqrt{21}}{2} \right )$ Question 33 :$Two\; adjacent\; sides\; of \; a \; parallelogram\; are\; \; \hat{i}-2\hat{j}-3\hat{k} \; and \; \; 2\hat{i}-4\hat{j}+5\hat{k}.$$Find \; the \; unit\; vector\; parallel\; to\; one \; of \; its \; diagonals.\; Also\; find \; its\; area.$$\; \; .....\; \; Ans\left ( \frac{1}{3}(3\hat{i}-6\hat{j} +2\hat{k}), \; 11\sqrt{5}\; sq.unit \right )$
Questions based on the scalar triple  product
 Question 34 : $Find\; \lambda \; so \; that\; the\; four \; points\; with \; position\; vectors \; 3\hat{i}+2\hat{j}+ \hat{k},\; 4\hat{i}+\lambda \hat{j}+\; 5\hat{k}$$4\hat{i}+2\hat{j}-2 \hat{k}\; and\; 6\hat{i}+5\hat{j}- \hat{k}\; are\; coplanar.\; \; ......\; \; Ans[\lambda =5]$ Question 35 : $Find\; the \; volume \; of \; the\; parallelopiped\; whose \; coterminous\; edges \; are$$\vec{a} =2\hat{i}-3\hat{j}+ 4\hat{k}, \; \vec{b}=\hat{i}+2\hat{j}-1\hat{k} \; and \; \vec{c}=2\hat{i}-\hat{j}+ 2\hat{k}\: \: .....\: \: Ans[2]$ Question 36 : $The \; volume \; of \; the\; parallelopiped\; whose \; coterminous\; edges \; are\; \vec{a}=-12\hat{i}+\lambda \hat{k}$$\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]$ 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}]$ 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,$$then\; find\; the\; value \; of\; \; pqr-(p+q+r)\; \; \; ......\; \; Ans[-2]$ Question 39 $Let\; \vec{a}=\hat{i}-\hat{j},\; \vec{b}=\hat{i}+\hat{j}+\hat{k} \; and\; \vec{c}\; be\; a \; vector\; such\; that\;\; \vec{a}\times \vec{c}+\vec{b}=\vec{0}$$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$$relations \; for \; the\; value \; of\; x,\; y,\; z,\; then\; find\; \vec{c} \; then \; its \; magnitude.$ Question 40 $If \; \; \vec{a}=\hat{i}+2\hat{j}+4\hat{k},\; \; \vec{b}=\hat{i}+\lambda \hat{j}+4\hat{k} \; \; and \; \; \vec{c}=2\hat{i}+4\hat{j}+(\lambda ^{2}-1)\hat{k}$$be\; copanar\; vectors.\; Then \; find\; the\; value \; of\; \vec{a}\times \vec{c}\; \; ....\; \; Ans[-10\hat{i}+5\hat{j}]$ 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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