Write the value of the product of vectors \[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\].
Last updated date: 28th Mar 2023
•
Total views: 308.4k
•
Views today: 7.89k
Answer
308.4k+ views
Hint: The cross product two unit vector is the vector perpendicular to both of them. And, the dot product of two unit vectors in the same direction is equal to 1.
We have , an equation \[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\]. The vectors i, j and k are the vectors in x-direction , y-direction and the z-direction.
There are three operations involved in these expressions.
1) Cross product of two unit vector
2) Dot product of two unit vector
3) Addition
Cross product of any two unit vector can be given by remembering small sequence
$
{\text{i}} \to {\text{j}} \to {\text{k}} \to {\text{i}} \\
{\text{i}} \times {\text{j = k}} \\
{\text{j}} \times {\text{k = i}} \\
{\text{k}} \times {\text{i = j}} \\
$
Applying it on the given equation, we have
\[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\]= k.k + i.i
Now, the dot product of two vectors is equal to the multiplication of length of the two vectors and cosine of the angle between them.
${\text{a}}{\text{.b = |a||b|cos}}\theta $ where $\theta $ is the angle between vectors a and b.
The length of the unit vector is 1. And if angle is 0 degree, then a.b = 1.
So,
\[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\]= k.k + i.i
=1 + 1
= 2
The value of \[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\]is 2.
Note: In these types of questions, we need to start with solving the brackets , then the whole expression. The cross product of two unit vectors in the same direction is zero. The dot product of two perpendicular unit vectors is zero.
We have , an equation \[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\]. The vectors i, j and k are the vectors in x-direction , y-direction and the z-direction.
There are three operations involved in these expressions.
1) Cross product of two unit vector
2) Dot product of two unit vector
3) Addition
Cross product of any two unit vector can be given by remembering small sequence
$
{\text{i}} \to {\text{j}} \to {\text{k}} \to {\text{i}} \\
{\text{i}} \times {\text{j = k}} \\
{\text{j}} \times {\text{k = i}} \\
{\text{k}} \times {\text{i = j}} \\
$
Applying it on the given equation, we have
\[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\]= k.k + i.i
Now, the dot product of two vectors is equal to the multiplication of length of the two vectors and cosine of the angle between them.
${\text{a}}{\text{.b = |a||b|cos}}\theta $ where $\theta $ is the angle between vectors a and b.
The length of the unit vector is 1. And if angle is 0 degree, then a.b = 1.
So,
\[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\]= k.k + i.i
=1 + 1
= 2
The value of \[\left( {{\text{i}} \times {\text{j}}} \right).{\text{k + }}\left( {{\text{j}} \times {\text{k}}} \right).{\text{i }}\]is 2.
Note: In these types of questions, we need to start with solving the brackets , then the whole expression. The cross product of two unit vectors in the same direction is zero. The dot product of two perpendicular unit vectors is zero.
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?
