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Given  and (î, ĵ and k̂ are unit vectors along x, y & z axes respectively). The unit vector in the direction of is:

Answer (Detailed Solution Below)

Option 3 :
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Detailed Solution

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Concept:

  • The unit vector û in the direction of a vector  is given by: , where  is the magnitude of the vector .
  • The magnitude of a vector  is given as: 

.

Calculation:

We have  and .

∴ 

⇒ 

The magnitude of  will be:

Now, the unit vector along the direction of  will be:

⇒ .

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Q1.Let ABCDEF be a regular hexagon. If \(\rm \vec{AD}=m \vec {BC}\ and\ \vec {CF}=n\vec {AB}\) then what is mn equal to 
Q2.If \(\vec{A} = 2\hat{i}+5\hat{j}\) & \(\vec{B} = 2\hat{i}-\hat{j}\) then the unit vector along the \(|\vec{A}+\vec{B} |\) is
Q3.Unit vector of \(\rm \vec r=3i+2j-2k\) will be:
Q4.If î‚ ĵ and k̂ are unit vectors along X, Y and Z axes respectively, then the product (î × ĵ) will be equal to: 
Q5.The unit vector perpendicular to the vectors î − ĵ and î + ĵ forming a right handed system is
Q6.If λî + 2λĵ + 2λk̂ is a unit vector, then the value of λ is:
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More Vector Algebra Questions

Q1.The vectors \(\rm \vec a, \vec b\ and\ \vec c\) are of the same length. If taken pairwise they form equal angles. If \(\rm \vec a=̂ i+̂ j\ and \ \vec b=̂ j+̂ k,\) then what can \(\vec c\) be equal to?  I. î + k̂  II. \(\rm \frac{-\hat i+4\hat j-\hat k}{3}\) Select the correct answer using the code given below. 
Q2.Let ABCDEF be a regular hexagon. If \(\rm \vec{AD}=m \vec {BC}\ and\ \vec {CF}=n\vec {AB}\) then what is mn equal to 
Q3.Let θ be the angle between two unit vectors \(\rm \vec a\ and\ \vec b. \ if\ \vec a+2\vec b\) is perpendicular to \(\rm 5\vec a-4\vec b\) then what is cos θ + cos 2θ equal to?
Q4.For what value of the angle between the vectors \(\rm \vec a\ and \ \vec b\) is the quantity \(\rm |\vec a\times \vec b|+\sqrt3|\vec a.\vec b|\) maximum? 
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Q6.A vector \(\rm \vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in {R}) \) lies in the plane of the vectors, \(\rm \vec{b}=\hat{i}+\hat{j}\) and \(\rm \vec{c}=\hat{i}-\hat{j}+4 \hat{k}\). If \(\rm \vec{a}\) bisects the angle between \(\rm \vec{b}\) and \(\rm \vec{c}\), then
Q7.The vector equation of the plane passing through the intersection of the planes \(\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1\) and \(\vec{r} \cdot(\hat{i}-2 \hat{j})=-2\) and the point (1, 0, 2) is:
Q8.Let ABC be a triangle such that \(\overrightarrow{B C}=\vec{a}, \overrightarrow{C A}=\vec{b}, \overrightarrow{A B}=\vec{c},|\vec{a}|=6 \sqrt{2},|\vec{b}|=2 \sqrt{3}\) and \(\vec{b} \cdot \vec{c}=12\) Consider the statements : \(\text { (S1): }|(\vec{a} \times \vec{b})+(\vec{c} \times \vec{b})|-|\vec{c}|=6(2 \sqrt{2}-1)\) \(\text { (S2): } \angle A C B=\cos ^{-1}\left(\sqrt{\frac{2}{3}}\right)\) Then
Q9.Let a unit vector which makes an angle of 60° with 2î + 2ĵ - k̂ and an angle of 45° with î - k̂ be \(\vec{C}\). Then \(\vec{C}+ \left(-\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k}\right)\) is:
Q10.If a̅, b̅, c̅, d̅ are the position vectors of the points A, B, C, D, respectively, such that no three of them are collinear and a̅ + c̅ = b̅ + d̅. then the quadrilateral ABCD is: 
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