Show that the lines
`vec( r)=(2hat(i) -3hat(k)) + lambda(hat(i) +2hat(j)+3hat(k)) " and " vec(r )=(3hat(i) +6hat(j)+3hat(k)) + mu(2hat(i) +3hat(j) +4hat(k))`
intersect.
Also find their point of intersection.
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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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If ` vec a= hat a= hat i- hat j+7 hat k` and ` vec b=5 hat j- hat j+lambda hat k` , then find the value of `lambda,` so that ` vec a+ vec b` and ` vec
Correct Answer - `lambda = pm 5`
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If `vec(a)=(2hat(i)-hat(j)+hat(k)), vec(b)=(hat(i)-3hat(j)-5hat(k)) and vec(c)=(3hat(i)-4hat(j)-hat(k))`, find `[vec(a)vec(b)vec(c)]` and interpret th
Correct Answer - 0, the given vectors are coplanar
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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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`vec(r ) =(hat(i)+2hat(j) -4hat(k)) + lambda(2hat(i) +3hat(j) +6hat(k))` `vec(r )=(3hat(i) +3hat(j) -5hat(k)) + mu (-2hat(i) +3hat(j) +8hat(k))`
Correct Answer - `(14sqrt(241))/(241)` units
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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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Let \( \vec{a}, \vec{b}, \vec{c} \) be unit vectors such that \( \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} \). Which one of the following is correct ? [IIT-2007] (a) \( \vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a}=\overrightarrow{0} \) (b) \( \overrightarrow{ a } \times \overrightarrow{ b }=\overrightarrow{ b } \times \overrightarrow{ c }=\overrightarrow{ c } \times \overrightarrow{ a } \neq \overrightarrow{0} \) (c) \( \overrightarrow{ a } \times \overrightarrow{ b }=\overrightarrow{ b } \times \overrightarrow{ c }=\overrightarrow{ a } \times \overrightarrow{ c } \neq \overrightarrow{0} \) (d) \( \vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a} \) are mutually \( \perp \)
Given that \(\vec a, \vec b\) and \(\vec c\)are unit vectors. i.e., |\(\vec a\)| = |\(\vec b\)| = |\(\vec c\)| = 1 And \(\vec a\) + \(\vec b\) + \(\vec c\) = \(\vec 0\) Then \((\vec a+\vec b+\vec c).(\vec a+\vec b+\vec c)=\vec 0.\vec...
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