Let the vectors \(\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k},\)
\(\overrightarrow{ b }=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}\) and
\(\vec{c}=t \hat{i}-t \hat{j}+\hat{k},\)\(t \in R\) be such that for
\(\alpha, \beta, \gamma \in R, \alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}=\overrightarrow{0} \Rightarrow \alpha=\beta=\gamma=0\)
Then, the set of all values of t is :
Let the vectors \(\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k},\)
\(\overrightarrow{ b }=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}\) and
\(\vec{c}=t \hat{i}-t \hat{j}+\hat{k},\)\(t \in R\) be such that for
\(\alpha, \beta, \gamma \in R, \alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}=\overrightarrow{0} \Rightarrow \alpha=\beta=\gamma=0\)
Then, the set of all values of t is :
- a non-empty finite set
- equal to $N$
- equal to $R -\{0\}$
- equal to $R$
The Correct Option is C
Solution and Explanation
The correct option is (C): equal to $R -\{0\}$
Top Questions on Vectors
- Number of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \), that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
- If the sum of two vectors is a unit vector, then the magnitude of their difference is:
- If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively are \( \frac{16}{11} (3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{11} (-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:
- Let \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) be vectors of magnitude 2, 3, and 4 respectively. If: - \( \mathbf{a} \) is perpendicular to \( (\mathbf{b} + \mathbf{c}) \), - \( \mathbf{b} \) is perpendicular to \( (\mathbf{c} + \mathbf{a}) \), - \( \mathbf{c} \) is perpendicular to \( (\mathbf{a} + \mathbf{b}) \), then the magnitude of \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) is equal to:
- Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{c} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to:
Questions Asked in JEE Main exam
Considering Bohr’s atomic model for hydrogen atom :
(A) the energy of H atom in ground state is same as energy of He+ ion in its first excited state.
(B) the energy of H atom in ground state is same as that for Li++ ion in its second excited state.
(C) the energy of H atom in its ground state is same as that of He+ ion for its ground state.
(D) the energy of He+ ion in its first excited state is same as that for Li++ ion in its ground state.- JEE Main - 2025
- Atomic Models
- In an adiabatic process, which of the following statements is true?
- JEE Main - 2025
- Thermodynamics
- To obtain the given truth table, the following logic gate should be placed at G
- JEE Main - 2025
- Logic gates
- Choose the correct logic circuit for the given truth table having inputs A and B.
- JEE Main - 2025
- Logic gates
A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is:
Concepts Used:
Vectors
The quantities having magnitude as well as direction are known as Vectors or Vector quantities. Vectors are the objects which are found in accumulated form in vector spaces accompanying two types of operations. These operations within the vector space include the addition of two vectors and multiplication of the vector with a scalar quantity. These operations can alter the proportions and order of the vector but the result still remains in the vector space. It is often recognized by symbols such as U ,V, and W
Representation of a Vector :
A line having an arrowhead is known as a directed line. A segment of the directed line has both direction and magnitude. This segment of the directed line is known as a vector. It is represented by a or commonly as AB. In this line segment AB, A is the starting point and B is the terminal point of the line.
Types of Vectors:
Here we will be discussing different types of vectors. There are commonly 10 different types of vectors frequently used in maths. The 10 types of vectors are:
- Zero vector
- Unit Vector
- Position Vector
- Co-initial Vector
- Like and Unlike Vectors
- Coplanar Vector
- Collinear Vector
- Equal Vector
- Displacement Vector
- Negative of a Vector