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

6.2.1.If a = 7i - 3j and b = i + 5j, then find the following vectors: (i) a+b (ii) a+3b (iii) 3a+(1/2)b (iv) b-a (v) 4b-5a (vi) (3/2)a-b

(i) vec a+vec b=(7+1)hat i+(-3+5)hat j=8hat i+2hat j

(ii) vec a+3vec b=(7+3)hat i+(-3+15)hat j=10hat i+12hat j

(iii) 3vec a+tfrac12vec b=left(21+tfrac12right)hat i+left(-9+tfrac52right)hat j=432hat i-132hat j

(iv) vec b-vec a=(1-7)hat i+(5-(-3))hat j=-6hat i+8hat j

(v) 4vec b-5vec a=(4-35)hat i+(20+15)hat j=-31hat i+35hat j

(vi) tfrac32vec a-vec b=left(212-1right)hat i+left(-tfrac92-5right)hat j=192hat i-192hat j

Result

(i) 8hat i+2hat j (ii) 10hat i+12hat j (iii) 432hat i-132hat j (iv) -6hat i+8hat j (v) -31hat i+35hat j (vi) 192hat i-192hat j

6.2.2.If a = 6i - j, b = i + 5j and c = 3i + 5j, then find the magnitudes of the following vectors: (i) b-c (ii) a-2b+c (iii) c-b-a

(i) vec b-vec c=(1-3)hat i+(5-5)hat j=-2hat i;quad |vec b-vec c|=sqrt{4}=2

(ii) vec a-2vec b+vec c=(6-2+3)hat i+(-1-10+5)hat j=7hat i-6hat j;quad |vec a-2vec b+vec c|=sqrt{49+36}=sqrt{85}

(iii) vec c-vec b-vec a=(3-1-6)hat i+(5-5+1)hat j=-4hat i+hat j;quad |vec c-vec b-vec a|=sqrt{16+1}=sqrt{17}

Result

(i) 2quad (ii) sqrt{85}quad (iii) sqrt{17}

6.2.3.Find the values of x and y in each of the following equations: (i) (xi+yj)+(2i+3j)=7i+6j (ii) (xi-5j)+(3i+5j)=-8i+yj (iii) (yi+3j)+(-5i+2xj)=9i+7j

(i) (x+2)hat i+(y+3)hat j=7hat i+6hat j ⇒ x+2=7, y+3=6 ⇒ x=5, y=3

(ii) (x+3)hat i+0hat j=-8hat i+yhat j ⇒ x+3=-8, 0=y ⇒ x=-11, y=0

(iii) (y-5)hat i+(3+2x)hat j=9hat i+7hat j ⇒ y-5=9, 3+2x=7 ⇒ y=14, x=2

Result

(i) x=5,y=3quad (ii) x=-11,y=0quad (iii) x=2,y=14

6.2.4.If a = i + 3j, c = 2i + j and a + 2b = c, then find |b|.

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

vec b=tfrac12hat i-hat j

Result

|vec b|=sqrt{left(tfrac12right)²+(-1)²}=sqrt{tfrac54}=sqrt52

6.2.5.If a = -2i + 7j and b = 3i - 5j, then find the vector components of a + 5b.

Working

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

=(-2+15)hat i+(7-25)hat j

Result

vec a+5vec b=13hat i-18hat j

6.2.6.If 5i - 3j = m(i - 10j) + n(4i - 3j), then find the values of m and n.

Expand RHS m(hat i-10hat j)+n(4hat i-3hat j)=(m+4n)hat i+(-10m-3n)hat j

Equate coefficients m+4n=5 ldots(1)qquad -10m-3n=-3 ldots(2)

Solve m=5-4n; -10(5-4n)-3n=-3 ⇒ 37n=47 ⇒ n=4737

m=5-4left(4737right)=185-18837=-337

Result

m=-337, n=4737

Unit 6: Vectors in Plane — Exercise 6 2