Mathematical problem is more important for creativity.

1. *** Two vector quantities modulus of each is 5 unit, are acting at a point
making an angle 120° with each other. Determine the magnitude and
direction of the resultant.
Given that,
P = Q = 5 Unit
θ= 120°
R = ?
Let R is the constant,
We know,
R = 𝑃² + 𝑄² + 2𝑃𝑄 cos 𝛼
𝑜𝑟, 𝑅 = 5 2 + 5 2 + 2.5.5 cos 120°
𝑜𝑟, 𝑅 = 25 + 25 + 50. (−
1
2
)
𝑜𝑟, 𝑅 = 50 − 25
𝑜𝑟, 𝑅 = 25
𝑜𝑟, 𝑅 = 5 unit.
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2. Again,
tan 𝜃 =
𝑄 sin 𝛼
𝑃 + 𝑄 cos 𝛼
or, tan 𝜃 =
5 sin 120°
5 + 5 cos 120°
Where 𝜃 is the angle between vector 𝑃 and resultant 𝑅 .
∴tan 𝜃 =
5 ×
3
2
5−
5
2
[∵ 𝑠𝑖𝑛 120° =
√3
2
, 𝑐𝑜𝑠 120° = −
1
2
]
𝑜𝑟, 𝑡𝑎𝑛 𝜃 =
5 × 3 × 2
5 × 2
𝑜𝑟, 𝑡𝑎𝑛 𝜃 = 3
𝑜𝑟, 𝑡𝑎𝑛 𝜃 = 𝑡𝑎𝑛 60°
∴ 𝜃 = 60°
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3. *** If 𝑨 = 𝟔𝒊 -𝟑𝒋 +𝟐𝒌 and, 𝑩= 𝟐𝒊+𝟐𝒋+𝒌, then find, 𝑨. 𝑩 and 𝑨 × 𝑩.
Given that,
𝐴 = 6𝑖 -3𝑗 +2𝑘
𝐵= 2𝑖+2𝑗+𝑘
𝐴. 𝐵 = (6𝑖 -3𝑗 +2𝑘). ( 2𝑖+2𝑗+𝑘)
= 12-6+2
= 8
𝐴 × 𝐵 =
𝑖 𝑗 𝑘
6 −3 2
2 2 1
= 𝑖(-3-4) - 𝑗(6-4)+ 𝑘(12+6)
= -7𝑖 -2𝑗 +18 𝑘 [Ans]
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4. ***If 𝑷 = 2𝒊 + 𝒋-𝟐𝒌 and 𝑸=𝟓𝒊-𝟒𝒋+𝒌, 𝒕𝒉𝒆𝒏 find the angle between 𝑷 𝒂𝒏𝒅 𝑸
We know,
𝑃.𝑄= PQ cos 𝜃
PQ cos 𝜃 = 𝑃.𝑄
or, cos 𝜃 =
𝑝 .𝑄 .
𝑃𝑄
----------(1)
Now,
𝑃.𝑄 = 2𝑖 + 𝑗 − 2𝑘) (5𝑖 − 4𝑗 + 𝑘
= 10 – 4 - 2
=4
𝑃 = 𝑃𝑥
2
+ 𝑃𝑦
2
+ 𝑃𝑧
2
or, P = 2 2 + 1 2 + −2 2
= 4 + 1 + 4
= 9
= 3
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5. 𝑄 = 𝑄𝑥
2
+ 𝑄𝑦
2
+ 𝑄𝑧
2
or, Q = 5 2 + −4 2 + 1 2
= 25 + 16 + 1
= 42
From equation (1)
cos 𝜃 =
𝑝 .𝑄
𝑃𝑄
or,cos 𝜃 =
4
3. 42
or, cos 𝜃 = 0.206
or, 𝜃= Cos -1(0.206)
or, 𝜃= 78º [Ans]
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