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Let veca and vecb be two non- zero perpe...

Let `veca and vecb` be two non- zero perpendicular vectors. A vector `vecr` satisfying the equation `vecr xx vecb = veca ` can be

A

`vecb- (vecaxx vecb)/(|vecb|^(2))`

B

`2 vecb - (veca xx vecb)/(|vecb|^(2))`

C

`|veca|vecb- (vecaxxvecb)/(|vecb|^(2))`

D

`|vecb|vecb- (veca xx vecb)/(|vecb|^(2))`

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To solve the problem, we need to find a vector \(\vec{r}\) that satisfies the equation \(\vec{r} \times \vec{b} = \vec{a}\), given that \(\vec{a}\) and \(\vec{b}\) are two non-zero perpendicular vectors. ### Step-by-Step Solution: 1. **Understand the Given Information**: - We know that \(\vec{a}\) and \(\vec{b}\) are perpendicular vectors. Therefore, the dot product \(\vec{a} \cdot \vec{b} = 0\). - The equation we need to satisfy is \(\vec{r} \times \vec{b} = \vec{a}\). 2. **Assume a General Form for \(\vec{r}\)**: - We can express \(\vec{r}\) as a linear combination of \(\vec{a}\), \(\vec{b}\), and \(\vec{a} \times \vec{b}\): \[ \vec{r} = x\vec{a} + y\vec{b} + z(\vec{a} \times \vec{b}) \] - Here, \(x\), \(y\), and \(z\) are scalars. 3. **Substitute \(\vec{r}\) into the Cross Product**: - Now, we substitute \(\vec{r}\) into the equation \(\vec{r} \times \vec{b}\): \[ \vec{r} \times \vec{b} = (x\vec{a} + y\vec{b} + z(\vec{a} \times \vec{b})) \times \vec{b} \] - Using the properties of the cross product, we have: \[ \vec{r} \times \vec{b} = x(\vec{a} \times \vec{b}) + y(\vec{b} \times \vec{b}) + z((\vec{a} \times \vec{b}) \times \vec{b}) \] - Since \(\vec{b} \times \vec{b} = \vec{0}\), this simplifies to: \[ \vec{r} \times \vec{b} = x(\vec{a} \times \vec{b}) + z((\vec{a} \times \vec{b}) \times \vec{b}) \] 4. **Use the Vector Triple Product Identity**: - The vector triple product identity states that: \[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w} \] - Applying this to \((\vec{a} \times \vec{b}) \times \vec{b}\): \[ (\vec{a} \times \vec{b}) \times \vec{b} = (\vec{a} \cdot \vec{b})\vec{b} - (\vec{b} \cdot \vec{b})\vec{a} \] - Since \(\vec{a} \cdot \vec{b} = 0\), this simplifies to: \[ (\vec{a} \times \vec{b}) \times \vec{b} = -|\vec{b}|^2 \vec{a} \] 5. **Substitute Back**: - Now substituting this back into our equation: \[ \vec{r} \times \vec{b} = x(\vec{a} \times \vec{b}) - z|\vec{b}|^2 \vec{a} \] - Setting this equal to \(\vec{a}\): \[ x(\vec{a} \times \vec{b}) - z|\vec{b}|^2 \vec{a} = \vec{a} \] 6. **Equate Coefficients**: - For the equation to hold, the coefficients of \(\vec{a}\) and \(\vec{a} \times \vec{b}\) must be equal to zero: - Coefficient of \(\vec{a}\): \(-z|\vec{b}|^2 = 1\) → \(z = -\frac{1}{|\vec{b}|^2}\) - Coefficient of \(\vec{a} \times \vec{b}\): \(x = 0\) 7. **Final Expression for \(\vec{r}\)**: - Substituting \(x = 0\) and \(z = -\frac{1}{|\vec{b}|^2}\) into the expression for \(\vec{r}\): \[ \vec{r} = 0 \cdot \vec{a} + y\vec{b} - \frac{1}{|\vec{b}|^2}(\vec{a} \times \vec{b}) \] - Thus, we have: \[ \vec{r} = y\vec{b} - \frac{1}{|\vec{b}|^2}(\vec{a} \times \vec{b}) \] ### Conclusion: The vector \(\vec{r}\) can be expressed as: \[ \vec{r} = y\vec{b} - \frac{1}{|\vec{b}|^2}(\vec{a} \times \vec{b}) \] where \(y\) can be any scalar value.

To solve the problem, we need to find a vector \(\vec{r}\) that satisfies the equation \(\vec{r} \times \vec{b} = \vec{a}\), given that \(\vec{a}\) and \(\vec{b}\) are two non-zero perpendicular vectors. ### Step-by-Step Solution: 1. **Understand the Given Information**: - We know that \(\vec{a}\) and \(\vec{b}\) are perpendicular vectors. Therefore, the dot product \(\vec{a} \cdot \vec{b} = 0\). - The equation we need to satisfy is \(\vec{r} \times \vec{b} = \vec{a}\). ...
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CENGAGE ENGLISH-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercises MCQ
  1. a(1), a(2),a(3) in R - {0} and a(1)+ a(2)cos2x+ a(3)sin^(2)x=0 " for ...

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  2. If veca and vecb are two vectors and angle between them is theta , the...

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  3. Let veca and vecb be two non- zero perpendicular vectors. A vector vec...

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  4. If vector vec b=(t a nalpha,-1,2sqrt(sinalpha//2))a n d vec c=(t a na...

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  5. Let vecr be a unit vector satisfying vecr xx veca = vecb, " where " |v...

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  6. If veca and vecb are unequal unit vectors such that (veca - vecb) xx[ ...

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  7. If veca and vecb are two unit vectors perpenicualar to each other and ...

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  8. If vectors veca and vecb are non collinear then veca/(|veca|)+vecb/(|v...

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  9. If veca and vecb are non - zero vectors such that |veca + vecb| = |vec...

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  10. Let veca vecb and vecc be non- zero vectors aned vecV(1) =veca xx (vec...

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  11. Vectors vecA and vecB satisfying the vector equation vecA+ vecB = vec...

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  12. A vector vecd is equally inclined to three vectors veca=hati-hatj+hatk...

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  13. Vectors perpendicular tohati-hatj-hatk and in the plane of hati+hatj+h...

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  14. If the sides vec(AB) of an equilateral triangle ABC lying in the xy-pl...

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  15. Let hata be a unit vector and hatb a non zero vector non parallel to v...

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  16. veca ,vecb and vecc are unimodular and coplanar. A unit vector vecd is...

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  17. If veca + 2 vecb + 3 vecc = vec0 " then " veca xx vecb + vecb xx vecc...

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  18. Let veca and vecb be two non-collinear unit vectors. If vecu=veca-(vec...

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  19. if veca xx vecb = vecc ,vecb xx vecc = veca , " where " vecc ne vec0 ...

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  20. Let veca, vecb, and vecc be three non- coplanar vectors and vecd be a ...

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