10th Class Mathematics Unit 6 Vectors in Plane Numericals with Solutions

6.1.1.Name the quadrant in which each point lies: (i) (4, 3) (ii) (5, -4) (iii) (-6, 2) (iv) (-4, -4)

Given

Points (i) (4,3)quad (ii) (5,-4)quad (iii) (-6,2)quad (iv) (-4,-4)

(i) x>0, y>0 ⇒ I^st Quadrant

(ii) x>0, y<0 ⇒ IV^th Quadrant

(iii) x<0, y>0 ⇒ II^nd Quadrant

(iv) x<0, y<0 ⇒ III^rd Quadrant

Result

(i) I^stquad (ii) IV^thquad (iii) II^ndquad (iv) III^rd

6.1.2.Plot the following points on the coordinate plane: (i) A(3, -3) (ii) B(-3, 3) (iii) C(5, 7) (iv) D(-2, -4)

Given

Points A(3,-3), B(-3,3), C(5,7), D(-2,-4)

Result

Ain QIV, Bin QII, Cin QI, Din QIII

6.1.3.Name the tail and tip of the following vectors: (i)-(vi) [diagram of six arrows labelled A→B, P→Q, R→S, M→N, A→B, D→C]

Given

Diagram Six arrows with labelled endpoints

Result

(i) Ato Bquad (ii) Pto Qquad (iii) Rto Squad (iv) Mto Nquad (v) Ato Bquad (vi) Dto C

6.1.4.Write the vector AB in the form of xi + yj: (i) A(1,-7), B(-2,4) (ii) A(8,9), B(12,3)

Formula

vec{AB}=(x_B-x_A)hat i+(y_B-y_A)hat j

Given

(i) A(1,-7), B(-2,4)

Working

vec{AB}=(-2-1)hat i+(4-(-7))hat j

Result

(i) vec{AB}=-3hat i+11hat j

Given

(ii) A(8,9), B(12,3)

Working

vec{AB}=(12-8)hat i+(3-9)hat j

Result

(ii) vec{AB}=4hat i-6hat j

6.1.5.Find the magnitude of the vector a: (i) a = -3i + 2j (ii) a = 4i - 3j (iii) a = (1/2)i + (3/2)j

Formula

|vec a|=sqrt{a₁²+a₂²}

(i) |vec a|=sqrt{(-3)²+2²}=sqrt{13}

(ii) |vec a|=sqrt{4²+(-3)²}=sqrt{25}=5

(iii) |vec a|=sqrt{left(tfrac12right)²+left(tfrac32right)²}=sqrt{104=tfrac{sqrt{10}{2}

Result

(i) sqrt{13}quad (ii) 5quad (iii) tfrac{sqrt{10}{2}

6.1.6.Find a unit vector in the direction of the vector given below: (i) a = -4i + 5j (ii) a = 6i + 8j (iii) a = sqrt(6) i + sqrt(6) j (iv) a = (1/2)i - (3/4)j

Formula

hat u=vec a|vec a|

(i) |vec a|=sqrt{16+25}=sqrt{41},quad vec u=dfrac{-4hat i+5hat j}{sqrt{41}

(ii) |vec a|=sqrt{36+64}=10,quad vec u=6hat i+8hat j10=3hat i+4hat j5

(iii) |vec a|=sqrt{6+6}=2sqrt3,quad vec u=sqrt6hat i+sqrt6hat j2sqrt3=sqrt2hat i+sqrt2hat j2

(iv) |vec a|=sqrt{tfrac14+916=dfrac{sqrt{13}{4},quad vec u=dfrac{2hat i-3hat j}{sqrt{13}

Result

(i) tfrac{-4hat i+5hat j}{sqrt{41} (ii) 3hat i+4hat j5 (iii) sqrt2hat i+sqrt2hat j2 (iv) tfrac{2hat i-3hat j}{sqrt{13}

6.1.7.If a = 5i - 7j, b = -i - j and c = 2i + 3j, then find unit vector parallel to a + b - 3c.

Working

vec a+vec b-3vec c=(5hat i-7hat j)+(-hat i-hat j)-3(2hat i+3hat j)

=(5-1-6)hat i+(-7-1-9)hat j=-2hat i-17hat j

vec v=-2hat i-17hat j,quad |vec v|=sqrt{4+289}=sqrt{293}

Result

vec u=dfrac{-2hat i-17hat j}{sqrt{293}

6.1.8.If a = 3i - j, b = -2i + 4j and c = i + 2j, then find unit vector parallel to 3a - 2c + 4b.

Working

3vec a-2vec c+4vec b=3(3hat i-hat j)-2(hat i+2hat j)+4(-2hat i+4hat j)

=(9-2-8)hat i+(-3-4+16)hat j=-hat i+9hat j

vec v=-hat i+9hat j,quad |vec v|=sqrt{1+81}=sqrt{82}

Result

vec u=dfrac{-hat i+9hat j}{sqrt{82}

6.1.9.Which of the following vectors are parallel? (i) a = 6i + j, b = 12i + 2j (ii) a = -2i + 3j, b = 6i - 9j (iii) a = 5i - 4j, b = 6i - 3j (iv) a = 3i - 7j, b = 6i - 14j

(i) vec b=2(6hat i+hat j)=2vec a ⇒ Parallel

(ii) vec b=-3(-2hat i+3hat j)=-3vec a ⇒ Parallel

(iii) a₁b₁=dfrac56, a₂b₂=-4-3=dfrac43; dfrac56nedfrac43 ⇒ Not parallel

(iv) vec b=2(3hat i-7hat j)=2vec a ⇒ Parallel

Result

Parallel pairs are (i), (ii) and (iv).

6.1.10.Find a vector thrice in length of 3i - 2j, but opposite in direction.

Given

vec v=3hat i-2hat j

3vec v=9hat i-6hat j

Opposite direction -3vec v=-(9hat i-6hat j)

Result

-9hat i+6hat j

6.1.11.Find two vectors that are double in magnitude of 3i - 5j, one in the same direction of it and other in its opposite direction.

Given

vec v=3hat i-5hat j

Same direction 2vec v=6hat i-10hat j

Opposite direction -2vec v=-6hat i+10hat j

Result

Same: 6hat i-10hat j; Opposite: -6hat i+10hat j

Unit 6: Vectors in Plane — Exercise 6 1