Ex 10.4, 3 - If a unit vector a makes angles pi/3 with i, pi/4 - Scalar product - Defination

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Ex 10.4, 3 If a unit vector 𝑎﷯ makes angles 𝜋﷮3﷯ with 𝑖﷯, 𝜋﷮4﷯ , with 𝑗﷯ and an acute angle θ with 𝑘﷯ , then find θ and hence, the components of 𝑎﷯ . Let us take a unit vector 𝑎﷯ = 𝑥 𝑖﷯ + y 𝑗﷯ + z 𝑘﷯ So, magnitude of 𝑎﷯ = 𝑎﷯﷯ = 1 Also, Angle of 𝑎﷯ with 𝑘﷯ = θ 𝑎﷯. 𝑘﷯ = 𝑎﷯﷯ 𝑘﷯﷯× cos﷮θ﷯ (x 𝑖﷯ + y 𝑗﷯ + z 𝑘﷯). (0 𝑖﷯ + .0 𝑗﷯ + 1 𝑘﷯) = 1 × 1 × cos θ (x × 0) + (y × 0) + (z × 1) = cosθ 0 + 0 + z = cos θ z = cos θ Now, Magnitude of 𝑎﷯ = ﷮𝑥2+𝑦2+𝑧2﷯ 1 = ﷮ 1﷮2﷯﷯﷮2﷯+ 1﷮ ﷮2﷯﷯﷯﷮2﷯+𝑐𝑜𝑠2θ﷯ 1 = ﷮ 1﷮4﷯+ 1﷮2﷯+𝑐𝑜𝑠2θ﷯ 1 = ﷮ 3﷮4﷯+𝑐𝑜𝑠2θ﷯ ﷮ 3﷮4﷯+𝑐𝑜𝑠2θ﷯ = 1 ﷮ 3﷮4﷯+𝑐𝑜𝑠2θ﷯﷯﷮2﷯ = 12 3﷮4﷯ + 𝑐𝑜𝑠2 θ = 1 𝑐𝑜𝑠2 θ = 1 − 3﷮4﷯ 𝑐𝑜𝑠2 θ = 1﷮4﷯ cos﷮θ﷯ = ± ﷮ 1﷮4﷯﷯ cos﷮θ﷯ = ± 1﷮2﷯ Since θ is given an acute angle So, θ < 90° ∴ θ is in 1st quadrant & cos θ is positive in 1st quadrant So, cos θ = + 1﷮2﷯ ∴ θ = 60° = 𝝅﷮𝟑﷯ Also, z = cos θ = cos 60° = 𝟏﷮𝟐﷯ Hence x = 1﷮2﷯ , y = 1﷮ ﷮2﷯﷯ & z = 1﷮2﷯ The required vector 𝑎﷯ is 1﷮2﷯ 𝑖﷯ + 1﷮ ﷮2﷯﷯ 𝑗﷯ + 1﷮2﷯ 𝑘﷯ So, components of 𝑎﷯ are 𝟏﷮𝟐﷯ , 𝟏﷮ ﷮𝟐﷯﷯ & 𝟏﷮𝟐﷯

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