704 + Interview Questions in VECTORS IN GENERAL AWARENESS Page 7 InterviewSolution

301.	If `hata_(1)` and `hata_(2)` are two non collinear unit vectors inclined at `60^(@)` to each other then the value of `(hata_(1)-hata_(2)).(2hata_(1)+hata_(2))` is
Answer» `a_(1)=a_(2)=1` `(veca_(1)-veca_(2)).(2veca_(1)+veca_(2))=2a_(1)^(2)-a_(2)^(2)-a_(1)a_(2)cos theta` `=2-1-(1)/(2)=(1)/(2)`

Discussion

302.	Find the angle between two vectors `hatA=2hati+hatj-hatk` and `hatB=hati-hatk`
Answer» `cos theta=(vecA,vecB)/(\|vecA\|\|vecB\|)=(2+0+1)/(sqrt(6)sqrt(2))=(1)/(2sqrt(2))sqrt(3)/(2)rArrtheta=30^(@)`

Discussion

303.	If `2hati-3hatj+4hatk` and `3hati+lambdahatj+muhatk` be collinear vectors, them find the values of `lambda` and `mu`.
Answer» Let `hatalpha=2hati-3hatj+4hatk` and `hatb=3hati+lambdahatj+muhatk` Given `hata & hatb` are collinear `thereforehataxxhatb=0:{:\|(hati,hatj,hatk),(2,-3,4),(3,lambda,mu):\|=0` `(-3mu-4lambda)hati-(2mu-12)hatj+(2lambda+9)hatk=0` Equating the coefficient of `hati,hatj` and `hatk` on both side we have `2mu-12=0 therefore mu=6` `2lambda+9=0rArr lambda=-(9)/(2)`

Discussion

304.	सदिश `overline(PQ)`, के अनुदिश मात्रक सदिश ज्ञात कीजिए जहाँ बिंदु P और Q क्रमश: (1,2,3) और (4,5,6) है!
Answer» The given points are `P(1,2,3) and Q(4, 5, 6)`. Therefore, `" "vec(PQ)=(4-1)hati+(5-2)hatj+(6-3)hatk=3hati+3hatj+3hatk` `" "\|vec(PQ)\|=sqrt(3^(2)+3^(2)+3^(2))=sqrt(9+9+9)=sqrt(27)=3sqrt(3)` Hence, the unit vector in the direction of `vec(PQ)` is `" "(vec(PQ))/(\|vec(PQ)\|)=(3hati+3hatj+3hatk)/(3sqrt(3))=(1)/(sqrt(3))hati+(1)/(sqrt(3))hatj+(1)/(sqrt(3))hatk`

Discussion

305.	If A = 5i + 6j + 9k; Find the unit vector of A.
Answer» Unit vector \(\vec A=\frac{\vec A}{\|\vec A\|}\) We have A = 5i + 6j + 9k \(\|\vec A\|=\sqrt{5^2+6^2+9^2}\) =\(\sqrt{142}\) Unit vector \(\vec A=\frac{5\hat i+6\hat j+9\hat k}{\sqrt{142}}\) \(=\frac{5}{\sqrt{142}}\hat i+\frac{6}{\sqrt{142}}\hat j+\frac{9}{\sqrt{142}}\hat k\)

305.

If A = 5i + 6j + 9k; Find the unit vector of A.

Answer»

Unit vector \(\vec A=\frac{\vec A}{|\vec A|}\)

We have

A = 5i + 6j + 9k

\(|\vec A|=\sqrt{5^2+6^2+9^2}\)

=\(\sqrt{142}\)

Unit vector \(\vec A=\frac{5\hat i+6\hat j+9\hat k}{\sqrt{142}}\)

\(=\frac{5}{\sqrt{142}}\hat i+\frac{6}{\sqrt{142}}\hat j+\frac{9}{\sqrt{142}}\hat k\)

Discussion

306.	An aeroplane is heading north east at a speed of `141.4ms^(-1)`. The north ward component of its velocty isA. `141.4ms^(-1)`B. `100ms^(-1)`C. zeroD. `50ms^(-1)`
Answer» Correct Answer - B `141.4sin 45^(@)`

Discussion

307.	What is the unit vector perpendicular to the following vectors` 2 hat(i) + 2 hat(j) - hat(k)` and `6 hat(i) - 3 hat(j) + 2 hat(k)`A. `(i + 10 j - 18 k)/(5 sqrt(17))`B. `(i - 10 j + 18 k)/(5 sqrt(17))`C. `(i - 10 j - 18 k)/(5 sqrt(17))`D. `(i + 10 j + 18 k)/(5 sqrt(17))`
Answer» Correct Answer - C `vec(A) = 2i + 2j - k , vec(B) = 6 i - 3 j + 2k` `vec(A) xx vec(B) = \|(I , j , k ) , (2 , 2 , -1), ( 6 , -3 , 2)\| = i - 10 j - 18 k` `\|vec(A) xx vec(B)\| = sqrt((1)^(2) + ( - 10)^(2) + (- 18)^(2)) = sqrt( 425) = 5 sqrt(17)` Unit vector along `(vec(A) xx vec(B))/(\|vec(A) xx vec(B)\|) = ( i - 10 j - 18 k)/( 5 sqrt(17))`

Discussion

308.	If `vecA=2hati+3hatj-hatk` and `vecB=-hati+3hatj+4hatk` then projection of `vecA` on `vecB` will beA. `3/(sqrt(13))`B. `3/(sqrt(26))`C. `sqrt(3/26)`D. `sqrt(3/13)`
Answer» Correct Answer - B

Discussion

309.	Two forces of `12N` and `8N` act upon a body. The resultant force on the body maximum value ofA. `4N`B. `0N`C. `20N`D. `8N`
Answer» Correct Answer - C

Discussion

310.	Find the angle that the vector `A = 2hati+3hatj-hatk` makes with y-axis.
Answer» Accoding to the resolution of the vector `costheta (A_(y))/(A)=(3)/(sqrt((2)^(2)+(3)^(2)+(-1)^(2)))=(3)/(sqrt(14))` `theta=cos^(-1)((3)/(sqrt(14)))`

Discussion

311.	Find the angle between two vector `A=2hati+hatj-hatk andB=hati-hatk.`
Answer» `a=\|A\|=sqrt((2)^(2)+(1)^(2)+(-1)^(2))=sqrt(6)` `B=\|b\|=sqrt((1)^(2)+(-1)^(2))=sqrt(2)` `A.B(2hati+hatj-hatk).(hati-hatk)` `Now , cos theta=(A.B)/(AB)=(3)/(sqrt(6).sqrt(2))=(3)/(sqrt(12))=(sqrt(3))/(2)` `thereforetheta=30^(@)`

Discussion

312.	prove that the vectors `A=2hati-3hatj+hatk and B=hati+hatj + hatk` are mutually perpendicular .
Answer» `A.B=(2hati-3hatj+hatk).(hati+hatj+hatk)` `=(2)(1)+(-3)(1)+(1)(1)` `=0=ABcos theta` `thereforecostheta=O (asAneO, bne O)` or the vectors A and Bare mutually perpendicular.

Discussion

313.	A velocity of `10 ms^(-1)` has its Y-component `5sqrt(2)ms^(-1)`. Calculate its X-component.
Answer» Correct Answer - `5sqrt(2)ms^(-1)`

Discussion

314.	`vecA` and `vecB` are two vectors gives by `vecA = 2 hati + 3hatj and vecB = hati +hatj`. The magnetiude of the component of `vecA` along `vecB` is :A. `(5)/(sqrt(5))`B. `(3)/(sqrt(2))`C. `(7)/(sqrt(2))`D. `(1)/(sqrt(2))`
Answer» Correct Answer - A `vecA .vecB = (2hati + 3hatj).((hati + hatj)/(sqrt(2)))` `=(2+3)/(sqrt(2))=(5)/(sqrt(2))`

Discussion

315.	The component of `vecA` along `vecB` is `sqrt(3)` times that of the component of `vecB` along `vecA`. Then A:B isA. `1:sqrt(3)`B. `sqrt(3):1`C. `2:sqrt(3)`D. `sqrt(3):2`
Answer» Correct Answer - B `A cos theta=(vecA.vecB)/(\|vecB\|)` and `B cos theta=(vecA.vecB)/(\|vecA\|)`, `A cos theta=sqrt(3) B cos theta`

Discussion

316.	Which of the following represents a unit vector ?A. `(\|A\|)/(A)`B. `(A)/(\|A\|)`C. `(A)/(A)`D. `(\|A\|)/(\|A\|)`
Answer» Correct Answer - B (b) A unit vector does is represented b`hatA` where `hatA=(A)/(\|A\|)`

Discussion

317.	Unit vector does not have anyA. directionB. magnitudeC. unitD. All of these
Answer» Correct Answer - C (C ) Unit vector does not have any unit.

Discussion

318.	Two forces 8 N and 12 act ay `120^(@)` The third force required to keep the body in equilbrium isA. 4 NB. `4 sqrt(7)N`C. 20 ND. none of these
Answer» Correct Answer - B (b) Third foece sh ould be equal and opposite to the resultant of the given vector .

Discussion

319.	Four co-planar concurrent forces are acting on a body as shown in the figure to keep it in equilibrium. The the values of P and `theta` are. A. `P=4N,theta=0^(@)`B. `P=2N,theta=90^(@)`C. `P=2N,theta=0^(@)`D. `P=4N,theta=90^(@)`
Answer» Correct Answer - B `P cos theta+sqrt(3)=2 sin 60^(@)` `P sin theta=1+2 cos 60^(@)`

Discussion

320.	A plane is travelling eastward at a speed of `500 kmh^(-1)`. But a `90 km h^(-1)` wind is blowing southward. What is the direction and speed of the plane relative to the ground ?
Answer» Correct Answer - `10.2^(@)` south of east, `508 kmh^(-1)`

Discussion

321.	The resultant of `vec(P)` and `vec(Q)` is `vec(R)`. If `vec(Q)` is doubled, `vec(R)` is doubled, when `vec(Q)` is reversed, `vec(R)` is again doubled. Find P:Q:R.
Answer» Let `theta` be the angle between `vec(P)`and `vec(Q)`.Then `R^(2)=\|vec(P)+vec(Q)\|^(2)=P^(2)+Q^(2)+2PQ cos theta`..(i) If `vec(Q)` is doubled, `vec(R )` is doubled. That means,the magnitude of resultant of `2vec(Q)` and `vec(P)` is `(2R)^(2)=P^(2)+(2Q)^(2)+2P(PQ)cos theta` This yields `4R^(2)=P^(2)+4Q^(2)+4PQ cos theta`..(ii) When `vec(Q)` is reversed, `vec(R )` is doubled. Hence, the magnitude of resultant of `vec(P)` and `(-vec(Q))` is 2R. Then `(2R)^(2)=P^(2)+Q^(2)+2PQ cos (180^(@)-theta)` This yields `4R^(2)=P^(2)+Q^(2)-2PQ cos theta`..(iii) Solving equation (i),(ii) and (iii) we obtain `Q=sqrt(3/2)R` and P=R`.

Discussion

322.	A boat takes two hours to travel 8 km and back in still water. If the velocity of water is 4 km/h, the time taken for going upstream 8 km and coming back isA. `2h`B. `2h 40` minC. `1h 20 `minD. cannot be estimated with the information given
Answer» Correct Answer - B

Discussion

323.	A man sitting in a bus travelling in a direction from west to east with a speed of `40km//h` observes that the rain-drops are falling vertically down. To the another man standing on ground the rain will appearA. To fall vertically downB. To fall at an angle going from west to eastC. To fall at an angle going from east to westD. The information given in insufficient to decide the direction of rain
Answer» Correct Answer - B

Discussion

324.	A man can swim with velocity v relative to water. He has to cross a river of width d flowing with a velocity `u (u gt v)`. The distance through which he is carried down stream by the river is x. Which of the following statements is correct?A. If he crosses the river in minimum time `x=(du)/v`B. `x` can not be less than `(du)/v`C. For `x` to be minimum he has to swim in a direction making an angle of `(pi)/2+"sin"^(-1)(v/u)` with the direction of the flow of waterD. `x` will be max. if he swims in a directin making an angle of `(pi)/2+sin^(-1)v/u` with direction of the flow of water
Answer» Correct Answer - A::C

Discussion

325.	Find the coordinates of the point which is located :In the YZ-plane, one unit to the right of the XZ-plane and six units above the XY-plane.
Answer» Let the coordinates of the point be (x, y, z). Since the point is located in the YZ plane, x = 0. Also, the point is one unit to the right of XZ-plane and six units above the XY-plane. ∴ y = 1, z = 6. Hence, coordinates of the required point are (0, 1, 6).

325.

Find the coordinates of the point which is located :In the YZ-plane, one unit to the right of the XZ-plane and six units above the XY-plane.

Answer»

Let the coordinates of the point be (x, y, z).

Since the point is located in the YZ plane, x = 0. Also, the point is one unit to the right of XZ-plane and six units above the XY-plane.

∴ y = 1, z = 6.

Hence, coordinates of the required point are (0, 1, 6).

Discussion

326.	Find the coordinates of the point which is located : Three units behind the YZ-plane, four units to the right of the XZ-plane and five units above the XY-plane.
Answer» Let the coordinates of the point be (x, y, z). Since the point is located 3 units behind the YZ- j plane, 4 units to the right of XZ-plane and 5 units , above the XY-plane, x = -3, y = 4 and z = 5 Hence, coordinates of the required point are (-3, 4, 5)

326.

Find the coordinates of the point which is located : Three units behind the YZ-plane, four units to the right of the XZ-plane and five units above the XY-plane.

Answer»

Let the coordinates of the point be (x, y, z). Since the point is located 3 units behind the YZ- j plane, 4 units to the right of XZ-plane and 5 units , above the XY-plane,

x = -3, y = 4 and z = 5

Hence, coordinates of the required point are (-3, 4, 5)

Discussion

327.	Evaluate:\( \:\int \:\left[\left(x^3-yz\right)dydz-2x^2ydzdx+zdxdy\right]\) over the surface of a cube bounded by the coordinates planes and the plane x = y = z = a.
Answer» We are finding double integration over the surface of a cube bounded by coordinate planes x = y = z = a and the plane x = y = z = a ∫∫[(x³ - yz) dy dz - 2x²y dz dx + zdxdy] surfaces are s₁ : x = 0, s₂ : x = a, s₃: y = 0, s₄: y = a, and s₅ : z = 0, s₆ : z = a \(\therefore\) ∫∫[(x³ - yz) dy dz - 2x²y dz dx + z dx dy] = (∫∫_s1 + ∫∫_s2 + ∫∫_s3 + ∫∫_s4 + ∫∫_s5 + ∫∫_s6) [(x³ - yz) dy dz - 2x²ydz dx + zdx dy] = \(\int\limits_{y=0}^{y=a}\int\limits_{z=0}^{z=a}(-yz)dydz\) + \(\int\limits_{y=0}^{y=a}\int\limits_{z=0}^{z=a}(a^3-yz)dydz\) + \(\int\limits_{z=0}^{z=a}\int\limits_{x=0}^{x=a}(-2x^2\times0)dzdx\) + \(\int\limits_{z=0}^{z=a}\int\limits_{x=0}^{x=a}(-2x^2a)dzdx\)+ \(\int\limits_{x=0}^{x=a}\int\limits_{y=0}^{y=a}0dxdy\) + \(\int\limits_{x=0}^{x=a}\int\limits_{y=0}^{y=a}adxdy\) = \(\int\limits_{y=0}^{y=a}-[\frac{yz^2}2]^{z=a}_{z=0}dy\) + \(\int\limits_{y=0}^{y=a}-[a^3z-\frac{yz^2}2]^{z=a}_{z=0}dy\) + 0 - 2a \(\int\limits_{z=0}^{z=a}-[\frac{x^3}3]^{x=a}_{x=0}dz\) + 0 + a\(\int\limits_{x=0}^{x=a}[y]^{y=a}_{y=0}dx\) = \(-\frac{a^2}2\int\limits_0^aydy+\int\limits_0^a(a^4-\frac{a^2y}2)dy\) - \(\frac{2a^4}3\int\limits_0^adz+a^2\int\limits_0^adx\) = \(-\frac{a^2}2[\frac{y^2}2]_0^a+[a^4y - \frac{a^2y^2}4]_0^4\) - \(\frac{2a^4}3[z]_0^a+a^2[x]_0^a\) = -\(\frac{a^4}4+a^5-\frac{a^4}4-\frac{2a^5}3+a^3\) = \(\frac13a^5-\frac12a^4+a^3\)

327.

Evaluate:\( \:\int \:\left[\left(x^3-yz\right)dydz-2x^2ydzdx+zdxdy\right]\) over the surface of a cube bounded by the coordinates planes and the plane x = y = z = a.

Answer»

We are finding double integration over the surface of a cube bounded by coordinate planes x = y = z = a and the plane x = y = z = a

∫∫[(x³ - yz) dy dz - 2x²y dz dx + zdxdy]

surfaces are s₁ : x = 0, s₂ : x = a, s₃: y = 0, s₄: y = a, and s₅ : z = 0, s₆ : z = a

\(\therefore\) ∫∫[(x³ - yz) dy dz - 2x²y dz dx + z dx dy]

= (∫∫_s1 + ∫∫_s2 + ∫∫_s3 + ∫∫_s4 + ∫∫_s5 + ∫∫_s6) [(x³ - yz) dy dz - 2x²ydz dx + zdx dy]

= \(\int\limits_{y=0}^{y=a}\int\limits_{z=0}^{z=a}(-yz)dydz\) + \(\int\limits_{y=0}^{y=a}\int\limits_{z=0}^{z=a}(a^3-yz)dydz\) + \(\int\limits_{z=0}^{z=a}\int\limits_{x=0}^{x=a}(-2x^2\times0)dzdx\)

+ \(\int\limits_{z=0}^{z=a}\int\limits_{x=0}^{x=a}(-2x^2a)dzdx\)+ \(\int\limits_{x=0}^{x=a}\int\limits_{y=0}^{y=a}0dxdy\) + \(\int\limits_{x=0}^{x=a}\int\limits_{y=0}^{y=a}adxdy\)

= \(\int\limits_{y=0}^{y=a}-[\frac{yz^2}2]^{z=a}_{z=0}dy\) + \(\int\limits_{y=0}^{y=a}-[a^3z-\frac{yz^2}2]^{z=a}_{z=0}dy\) + 0 - 2a \(\int\limits_{z=0}^{z=a}-[\frac{x^3}3]^{x=a}_{x=0}dz\) + 0 + a\(\int\limits_{x=0}^{x=a}[y]^{y=a}_{y=0}dx\)

= \(-\frac{a^2}2\int\limits_0^aydy+\int\limits_0^a(a^4-\frac{a^2y}2)dy\) - \(\frac{2a^4}3\int\limits_0^adz+a^2\int\limits_0^adx\)

= \(-\frac{a^2}2[\frac{y^2}2]_0^a+[a^4y - \frac{a^2y^2}4]_0^4\) - \(\frac{2a^4}3[z]_0^a+a^2[x]_0^a\)

= -\(\frac{a^4}4+a^5-\frac{a^4}4-\frac{2a^5}3+a^3\)

= \(\frac13a^5-\frac12a^4+a^3\)

Discussion

328.	The angle between two vectors `vec(A)= 3hat(i)+4hat(j)+5hat(k)` and `vec(B)= 3hat(i)+4hat(j)+5hat(k)` isA. `60^(@)`B. ZeroC. `90^(@)`D. None of these
Answer» Correct Answer - D `cos theta = (vec(A).vec(B))/(\|A\|\|B\|)= (9+16+25)/(sqrt(9+16+25)sqrt(9+16+25))` `=(50)/(50)= 1implies cos theta = 1 :. theta = cos^(-1)(1)= zero`

Discussion

329.	Which of the following vectors is//are perpendicular to the vector `4 I - 3 j` ?A. ` 4 i + 3j`B. `6 i`C. `7 k`D. `3 i - 4j`
Answer» Correct Answer - C `( 4 I - 3 j) . ( 7 k) = 0`

Discussion

330.	If a vector `2 hat (i) + 3 hat(j) + 8 hat(k)` is perpendicular to the vector `- 4 hat(i) + 4 hat(j) + alpha(k)`. Then the value of `alpha` isA. `-1`B. `-(1)/(2)`C. `(1)/(2)`D. `1`
Answer» Correct Answer - B `( 2 hat(i) + 3 hat(j) + 8 hat(k)). (-4 hat(i) + 4 hat(j) + alpha hat(k)) = 0` ` - 8 + 12 + 8 alpha = 0 rArr alpha = -(1)/(2)`

Discussion

331.	The angle between two vectors `vec(A)= 3hat(i)+4hat(j)+5hat(k)` and `vec(B)= 3hat(i)+4hat(j)+5hat(k)` isA. `90^(@)`B. `0^(@)`C. `60^(@)`D. `45^(@)`
Answer» Correct Answer - A `cos theta = (vec(A).vec(B))/(\|A\|\|B\|)= ((3hat(i)+4hat(j)+5hat(k))(3hat(i)+4hat(j)-5hat(k)))/(sqrt(9+16+25)sqrt(9+16+25))` `(9+16-25)/(50)=0` `implies cos theta= 0, :. theta= 90^(@)`

Discussion

332.	Of the following the vector quantity isA. TimeB. Electric CurrentC. Velocity of lightD. Gravitational force
Answer» Correct Answer - D

Discussion

333.	Of the following the scalar quantity isA. TemperatureB. Moment of forceC. Moment of coupleD. Magnetic moment
Answer» Correct Answer - A

Discussion

334.	Choose the correct statementA. Temperature is a scalar but temperature gradient is a vectorB. Velocity of a body is a vector but velocity of light is a scalarC. Electric intensity and Electric current density are vectorsD. All the above
Answer» Correct Answer - D

Discussion

335.	Choose the correct statementA. Scalar+vector=scalar/vectorB. `("vector")/("vector")="scalar"`C. Scalar/vector=scalar (Or) vectorD. vector-vector=vector.
Answer» Correct Answer - D

Discussion

336.	If `vecAxxvecB=vecC`, then which of the followig statements is wrongA. `vecC_\|_vecA`B. `vecA_\|_vecB`C. `vecC_\|_(vecA+vecB)`D. `vecC_\|_(vecA+vecB)`
Answer» Correct Answer - D

Discussion

337.	Choose the false statement:A. Electric current is a vector because it has both magnitude and directionsB. Timke is a vector which has direction always in the forward direction.C. All quantities having magnitude and direction are vector quantitiesD. All the above
Answer» Correct Answer - D

Discussion

338.	Which of the gollowing units could be associated with a vector quantity?A. newton/metreB. newton metre/secondC. kg `m^(2)` `s^(-2)`D. newton second
Answer» Correct Answer - D

Discussion

339.	Which of the following is meaningful?A. vector/vectorB. scalar/VectorC. Scalar+VectorD. Vector/Scalar
Answer» Correct Answer - D

Discussion

340.	Two forces each equal to `F//2` act at right angle. Their effect may be neutralized by a third force acting along their bisector in the opposite direction. What is the magnitude of that third forces.
Answer» Rsultant of two forces:`sqrt((F//2^(2))+(F//2^(2)))=F//sqrt(2)` The third force, should be opposite to the resultant and of same magnitude. Hence, answer is `F//sqrt(2)`.

Discussion

341.	A vector is not changed ifA. it is rotated through an arbitarary angleB. it is multiplied by an arbitarary scalarC. it is cross multiplied by a unit vectorD. it slides parallel to itslef.
Answer» Correct Answer - D

Discussion

342.	A vector is not changed ifA. it is rotated through an arbitrary angleB. it is multiplied by an arbitrary scalarC. it is cross multiplied by a unit vectorD. it is displaced paralled to itself
Answer» Correct Answer - D

Discussion

343.	If for two vectors `vec(A)` and `vec(B)`, sum `(vec(A)+vec(B))` is perpendicular to the difference `(vec(A)-vec(B))`. Find the ratio of their magnitude.A. `1`B. `2`C. `3`D. None of these
Answer» Correct Answer - A `(vec(A) + vec(B)). (vec(A) - vec(B)) = 0` `vec(A) . vec(A) - vec(A) . vec(B) + vec(B) . vec(A) - vec(B) . vec(B) = 0` `A^(2) - B^(2) = 0 rArr A = B` `(A)/(B) = 1`

Discussion

344.	If for two vectors `vec(A)` and `vec(B)`, sum `(vec(A)+vec(B))` is perpendicular to the difference `(vec(A)-vec(B))`. Find the ratio of their magnitude.A. 1B. 2C. 3D. None of these
Answer» Correct Answer - A `(vec(A)+vec(B))` is perpendicular to `(vec(A)-vec(B))`. Thus `(vec(A)+vec(B)).(vec(A)-vec(B))=0` or `A^(2)+vec(B).vec(A)-vec(A).vec(B)-B^(2)=0` Because of communative property of dot product `vec(A).vec(B)= vec(B).vec(A)` `:. A^(2)-B^(2)=0 or A= B` Thus the ratio of magnitudes `A//B= 1`

Discussion

345.	A vector is not changed ifA. it is displaced parallel to itselfB. it is roated through an arbitrary angleC. it is cross-multiplied by a unit vector.D. it is multiplied by an arbitary sacalr.
Answer» Correct Answer - A When a vector is displaced parallel to itself, neither its magnitude nor its direction changes.

Discussion

346.	The two vectors `vecA` and `vecB` are drawn from a common point and `vecC = vecA + vecB`. In column- I are given the conditions regarding the magnitudes of `vecA, vecB and vecC` as A, B, C respectively. Column- II gives the angle between the vectors `vecA and vecB`. Match them.
Answer» Correct Answer - `a to r; b to p; c to q,s; d to p` (a) `A^(2) + B^(2) = C^(2) rArr theta = 90^(@)` (b) `A^(2) + B^(2) gt C^(2) rArr theta = 90^(@)` `A^(2) + B^(2) lt C^(2) rArr theta = 90^(@)` (d) `A^(2) = B^(2) = C^(2) rArr theta = 120^(@) rArr theta gt 90^(@)`

Discussion

347.	Which one of the following statements is false regarding the vectors ?A. The magnitude of a vector is always a scalar.B. Each component of a vector is always a scalar.C. Two vector having different magnitudes cannot have their resultant Zero.D. Vectors obey triangle law of addition.
Answer» Correct Answer - B The magnitude of a vector is a pure number. Each component of a vector is also a vector. The resultant to two vectors can be zero only if they have the same magnitude and opposite directions.

Discussion

348.	Two vectors `vec(a)` and `vec(b)` are at an angle of `60^(@)` with each other . Their resultant makes an angle of `45^(@)` with `vec(a)` If `\|vec(b)\|=2`unit , then `\|vec(a)\|` isA. `sqrt(3)`B. `sqrt(3)-1`C. `sqrt(3)+1`D. `sqrt(3)//2`
Answer» Correct Answer - B `tan 45^(@)=(2sin 60^(@))/(a+2 cos 60^(@))=sqrt(3)/(a+1)` or `1=sqrt(3)/(a+1)` or `a+1=sqrt(3)` or `a=sqrt(3)-1`

Discussion

349.	Which one of the following statements is true?A. A scalar quantity is the one of that is conserved in a processB. A scalar quantity is the one of that can never take negative valuesC. A scalar quantity is the one that does not vary from one point to another in spaceD. A scalar quantity has the same value for observes with different orientation of the axes
Answer» Correct Answer - D A scalar quantity is independent of direction hence has the same value for observers with different orientation of the axis.

Discussion

350.	Two vectors `vec(A) and vec(B)` inclined at an angle `theta` have a resultant `vec(R )` which makes an angle `alpha` with `vec(A)`. If the directions of `vec(A) and vec(B)` are interchanged, the resultant will have the sameA. directionB. magnitudeC. direction as well as magnitudeD. None of these
Answer» Correct Answer - B Neither the magnitude of vector nor the angle between the vector is changed. So magnitude of the resultant remains unchanged. Howerver the direction of the resultatnt will be changed.

Discussion

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