Mathematics Multiple Choice Questions on “Addition of Vectors”.
1. If (vec{a})=(hat{i})+4(hat{j}) and (vec{b})=3(hat{i})-3(hat{j}). Find the magnitude of (vec{a}+vec{b}).
a) (sqrt{6})
b) (sqrt{11})
c) (sqrt{5})
d) (sqrt{17})
Answer: d
Clarification: Given that, (vec{a})=(hat{i})+4(hat{j}) and (vec{b})=3(hat{i})-3(hat{j})
∴(vec{a}+vec{b})=(1+3) (hat{i})+(4-3) (hat{j})
=4(hat{i})+(hat{j})
|(vec{a}+vec{b})|=(sqrt{4^2+1^2}=sqrt{16+1}=sqrt{17})
2. Find the sum of the vectors (vec{a})=6(hat{i})-3(hat{j}) and (vec{b})=5(hat{i})+4(hat{j}).
a) 11(hat{i})+(hat{j})
b) 11(hat{i})–(hat{j})
c) -11(hat{i})+(hat{j})
d) (hat{i})+(hat{j})
Answer: a
Clarification: Given that, (vec{a})=6(hat{i})-3(hat{j}) and (vec{b})=5(hat{i})+4(hat{j})
The sum of the vectors is given by (vec{a}+vec{b}).
∴(vec{a}+vec{b})=(6(hat{i})-3(hat{i}))+(5(hat{i})+4(hat{j}))
=(6+5) (hat{i})+(-3+4)(hat{j})
=11(hat{i})+(hat{j})
3. Find vector (vec{c}), if (vec{a})–(vec{b})+(vec{c})=6(hat{i})+8(hat{j}) where (vec{a})=7(hat{i})+2(hat{j}) and (vec{b})=4(hat{i})-5(hat{j}).
a) -3(hat{i})+(hat{j})
b) 3(hat{i})+(hat{j})
c) 3(hat{i})–(hat{j})
d) -3(hat{i})–(hat{j})
Answer: b
Clarification: Given that, (vec{a})–(vec{b})+(vec{c})=6(hat{i})+8(hat{j}) -(1)
It is also given that, (vec{a})=7(hat{i})+2(hat{j}) and (vec{b})=4(hat{i})-5(hat{j})
Substituting the values of (vec{a}) and (vec{b}) in equation (1), we get
(vec{a})–(vec{b})+(vec{c})=6(hat{i})+8(hat{j})
(7(hat{i})+2(hat{j}))-(4(hat{i})-5(hat{j}))+(vec{c})=6(hat{i})+8(hat{j})
∴(vec{c})=(6(hat{i})+8(hat{j}))-(7(hat{i})+2(hat{j}))+(4(hat{i})-5(hat{j}))
=(6-7+4) (hat{i})+(8-2-5) (hat{j})
=3(hat{i})+(hat{j})
4. Find the unit vector in the direction of the sum of the vectors, (vec{a})=2(hat{i})+7(hat{j}) and (vec{b})=(hat{i})-9(hat{j}).
a) (frac{3}{sqrt{11}} hat{i}-frac{2}{sqrt{11}} hat{j})
b) (frac{2}{sqrt{13}} hat{i}-frac{3}{sqrt{13}} hat{j})
c) –(frac{3}{sqrt{11}} hat{i}+frac{2}{sqrt{13}} hat{j})
d) (frac{3}{sqrt{13}} hat{i}-frac{2}{sqrt{13}} hat{j})
Answer: d
Clarification: Given that, (vec{a})=2(hat{i})+7(hat{j}) and (vec{b})=(hat{i})-9(hat{j})
The sum of the two vectors will be
(vec{a}+vec{b})=(2(hat{i})+7(hat{j}))+((hat{i})-9(hat{j}))
=(2+1) (hat{i})+(7-9)(hat{j})
=3(hat{i})-2(hat{j})
The unit vector in the direction of the sum of the vectors is
(frac{1}{|vec{a}+vec{b}|} (vec{a}+vec{b})=frac{3hat{i}-2hat{j}}{sqrt{3^2+(-2)^2}}=frac{3hat{i}-2hat{j}}{sqrt{13}}=frac{3}{1sqrt{3}} hat{i}-frac{2}{sqrt{13}}hat{j})
5. If (vec{a})=3(hat{i})+2(hat{j})+2(hat{k}), (vec{b})=2(hat{i})-8(hat{j})+(hat{k}), find (vec{a}+vec{b}).
a) 5(hat{i})+(hat{j})+3(hat{k})
b) 5(hat{i})-6(hat{j})+3(hat{k})
c) 5(hat{i})-6(hat{j})-3(hat{k})
d) 5(hat{i})+6(hat{j})+3(hat{k})
Answer: b
Clarification: It is given that, (vec{a})=3(hat{i})+2(hat{j})+2(hat{k}), (vec{b})=2(hat{i})-8(hat{j})+(hat{k})
To find: (vec{a}+vec{b})
∴(vec{a}+vec{b})=(3(hat{i})+2(hat{j})+2(hat{k}))+(2(hat{i})-8(hat{j})+(hat{k}))
=(3+2) (hat{i})+(2-8) (hat{j})+(2+1)(hat{k})
=5(hat{i})-6(hat{j})+3(hat{k})
6. Find the value of (vec{a}+vec{b})+(vec{c}), if (vec{a})=4(hat{i})-4(hat{j}), (vec{b})=-3(hat{i})+2k, (vec{c})=7(hat{j})-8(hat{k}).
a) (hat{i})-3(hat{j})
b) (hat{i})+3(hat{j})-6(hat{k})
c) (hat{i})+(hat{j})+6(hat{k})
d) (hat{i})+6(hat{k})
Answer: b
Clarification: Given that, (vec{a})=4(hat{i})-4(hat{j}), (vec{b})=-3(hat{i})+2k, (vec{c})=7(hat{j})-8(hat{k})
To find: (vec{a}+vec{b})+(vec{c})
∴(vec{a}+vec{b})+(vec{c})=(4(hat{i})-4(hat{j})) +(-3(hat{i})+2k) +(7(hat{j})-8(hat{k}))
=(4-3) (hat{i})+(-4+7) (hat{j})+(2-8)(hat{k})
=(hat{i})+3(hat{j})-6(hat{k})
7. Find the magnitude of (vec{a}+vec{b}), if (vec{a})=4(hat{i})+9(hat{j}) and (vec{b})=6(hat{i}).
a) (sqrt{181})
b) (sqrt{81})
c) (sqrt{11})
d) (sqrt{60})
Answer: a
Clarification: Given that, (vec{a})=4(hat{i})+9(hat{j}) and (vec{b})=6(hat{i})
∴(vec{a}+vec{b})=(4+6) (hat{i})+9(hat{j})
=10(hat{i})+9(hat{j})
|(vec{a}+vec{b})|=(sqrt{10^2+9^2}=sqrt{100+81}=sqrt{181})
8. Find vector (vec{b}), if (vec{a}+vec{b})+(vec{c})=8(hat{i})+2(hat{j}) where (vec{a})=(hat{i})-6(hat{j}) and (vec{c})=3(hat{i})+7(hat{j}).
a) 4(hat{i})+4(hat{j})
b) (hat{i})+4(hat{j})
c) 4(hat{i})–(hat{j})
d) 4(hat{i})+(hat{j})
Answer: d
Clarification: Given that, (vec{a}+vec{b})+(vec{c})=8(hat{i})+2(hat{j}) -(1)
Given: (vec{a})=(hat{i})-6(hat{j}) and (vec{c})=3(hat{i})+7(hat{j})
Substituting the values of (vec{a}) and (vec{b}) in equation (1), we get
(vec{a}+vec{b})+(vec{c})=8(hat{i})+2(hat{j})
((hat{i})-6(hat{j}))+(vec{b})+(3(hat{i})+7(hat{j}))=8(hat{i})+2(hat{j})
∴(vec{c})=(8(hat{i})+2(hat{j}))-((hat{i})-6(hat{j}))-(3(hat{i})+7(hat{j}))
=(8-1-3) (hat{i})+(2+6-7) (hat{j})
=4(hat{i})+(hat{j})
9. If (vec{a})=9(hat{i})-2(hat{j})+7(hat{k}), (vec{b})=5(hat{i})+(hat{j})-3(hat{k}), find (vec{a}+vec{b}).
a) (hat{i})–(hat{j})+4(hat{k})
b) 14(hat{i})–(hat{j})+4(hat{k})
c) 14(hat{i})-3(hat{j})+4(hat{k})
d) 14(hat{i})–(hat{j})+9(hat{k})
Answer: b
Clarification: Given that, (vec{a})=9(hat{i})-2(hat{j})+7(hat{k}), (vec{b})=5(hat{i})+(hat{j})-3(hat{k})
We have to find (vec{a}+vec{b})
∴(vec{a}+vec{b})=(9(hat{i})-2(hat{j})+7(hat{k}))+(5(hat{i})+(hat{j})-3(hat{k}))
=(9+5) (hat{i})+(-2+1) (hat{j})+(7-3)(hat{k})
=14(hat{i})–(hat{j})+4(hat{k})
10. Find the sum of the vectors (vec{a})=8(hat{i})+5(hat{j}) and (vec{b})=-2(hat{i})+6(hat{j})
a) 6(hat{i})+(hat{j})
b) 6(hat{i})+11(hat{j})
c) 6(hat{i})-11(hat{j})
d) (hat{i})+11(hat{j})
Answer: b
Clarification: Given that, (vec{a})=8(hat{i})+5(hat{j}) and (vec{b})=-2(hat{i})+6(hat{j})
∴The sum of the vectors will be
(vec{a}+vec{b})=(8(hat{i})+5(hat{j}))+(-2(hat{i})+6(hat{j}))
=(8-2) (hat{i})+(5+6)(hat{j})
=6(hat{i})+11(hat{j})