Question

A and b are two vectors with magnitude of 20 and 25 respective.the angle between them is 37°.the angle between a+b and a will be?

Answer 1

|a| = 20 , |b| = 25 and angle between a and b = 37°, e.g., Ф = 37°
Use formula, |a + b| = √{a² + b² + 2abcosФ}
= √{20² + 25² + 2 × 20 × 25cos37°}
= √{400 + 625 + 1000 × 4/5 }
= √{400 + 625 + 800} = √1825 = 5√73 -----(1)

Now, let angle between (a + b) and a is α
Use formula, cosα = (a + b).a/|a + b|.|a|
cosα = (a² + a.b)/5√73 × 20 [ from equation (1)]
We know, a.b = |a||b|cosФ = 20 × 25 × 4/5 = 400
So, cosα = (20² + 400)/100√73
cosα = 800/100√73 = 8/√73
Hence, α = cos⁻¹(8/√73)

Answer 2

| एक | = 20, | बी | = 25 और ए और बी = 37 डिग्री के बीच कोण, उदाहरण के लिए, Ф = 37 डिग्री
सूत्र का प्रयोग करें, | ए + बी | = √ {ए² + बी² + 2abcosФ}
= √ {20 वर्ग + 25 वर्ग + 2 × 20 × 25cos37 °}
= √ {400 + 625 + 1000 × 4/5}
= √ {400 + 625 + 800} = √1825 = 5√73 ----- (1)

अब, कोण (ए + बी) और ए के बीच कोण दें
सूत्र का प्रयोग करें, cosα = (ए + बी)। ए / | ए + बी |। | ए |
cosα = (a² + a.b) / 5√73 × 20 [समीकरण से (1)]
हम जानते हैं, ए.बी = | ए || बी | cosФ = 20 × 25 × 4/5 = 400
तो, cosα = (20² + 400) / 100√73
cosα = 800 / 100√73 = 8 / √73
इसलिए, α = cos⁻¹ (8 / √73)