A line is drawn in the direction of `(hat(i) +hat(j) -2hat(k))` and it passes through a point with position vector `(2hat(i) -hat(j) - 4hat(k))` .Find the equations of the line in the vector as well as Cartesian forms.
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Find `vec(A).vec(b)` when (i) `vec(a)=hat(i)-2hat(j)+hat(k)` and `vec(b)=3 hat(i)-4 hat(j)-2 hat(k)` (ii) `vec(a)=hat(i)+2hat(j)+3hat(k)` and `vec(b)=
Correct Answer - (i) 9 (ii) 8 (iii) -7
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Find the value of `lambda` for which the four points with position vectors `(-hat(j)+hat(k)), (2hat(i)-hat(j)-hat(k)), (hat(i)+lambda hat(j)+hat(k))an
Correct Answer - `lambda=1`
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A line passes through the point (2,1,-3) and is parallel to the vector `(hat(i) -2hat(j) +2hat(k))` . Find the equations of the line in vector and Car
Correct Answer - `vec(r )=(2hat(i)-hat(j)-3hat(k)) +lambda (hat(i) -2hat(j)+3hat(k)) ,(x-2)/(1)=(y-1)/(-2)=(z+3)/(3)`
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Find the vector equations of the line passing through the point with position vector `(2hat(i) +hat(j) -5hat(k))` and parallel to the vector `(hat(i)
Correct Answer - `vec(r ) =(2hat(i) +hat(j) -5hat(k)) +lambda(hat(i) +3hat(j)-hat(k)) ,(x-2)/(1)=(y-1)/(3)=(z+5)/(-1)`
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`vec(r ) =(3hat(i) +hat(j)-2hat(k)) +lambda (hat(i)-hat(j)-2hat(k)) " and " vec(r ) =(2hat(i) -hat(j) -5hat(k)) + mu (3hat(i) -5hat(j) -4hat(k))` FInd
Correct Answer - `cos^(-1) .((8sqrt(3))/(15))`
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`vec(r )=(hat(i) +hat(j)) +lambda(2hat(i) -hat(j) +hat(k))` `vec(r )=(2hat(i) +hat(j) -hat(k)) + mu (3hat(i) -5hat(j) +2hat(k))`
Correct Answer - `(10)/(sqrt(59))` units
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`vec(r ) =(hat(i) +2hat(j) +3hat(k)) + lambda(hat(i) -3hat(j) +2hat(k))` `vec(r ) =(4hat(i) +5hat(j) +6hat(k)) + mu (2hat(i) + hat(j) +2hat(k))` Find
Correct Answer - `(3sqrt(19))/(19)` units
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`vec(r )=(hat(i) +2hat(j) +hat(k)) + lambda(hat(i)-hat(j) + hat(k))` `vec(r )=(2hat(i) -hat(j) -hat(k)) + lambda(2hat(i) + hat(j) +2hat(k))`
Correct Answer - `(3sqrt(2))/(2)` units
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If `vec(a)=(4hat(i)+ 5 hat(j) - hat(k)),vec(b)=(hat(i)-4 hat(j)+ 5 hat(k))`, and`vec(c)=(3 hat(i)+ hat(j)-hat(k))` find a vector `vec(d)` which is per
Correct Answer - `vec(d)=7(hat(i)-hat(j)-hat(k))`
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12 In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by \( \hat{ k } \) and \( 2 \hat{ i }-2 \hat{ j } \), respectively. What is the unit vector along direction of propagation of the wave? (a) \( \frac{1}{\sqrt{2}}(\hat{ i }+\hat{ j }) \) (b) \( \frac{1}{\sqrt{2}}(\hat{ j }+\hat{ k }) \) (c) \( \frac{1}{\sqrt{5}}(\hat{ i }+2 \hat{ j }) \) (d) \( \frac{1}{\sqrt{5}}(2 \hat{ i }+\hat{ j }) \)
Correct answer is (a) We know that propagation wave vector ∵ \(\vec{E} = \hat{k}\) \(\vec{B} = 2\hat{i} - 2\hat{j}\) \(\vec{C} = \vec{E} \times \vec{B}\) \(\vec{C} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1...
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