If [latex]\vec{A} = 4\hat{i}-\hat{j}+3\hat{k}[/latex] and [latex]\vec{B}=7\hat{i}+5\hat{j}+\hat{k}[/latex], find the angle between the vectors [latex]\vec{A}[/latex] and [latex]\vec{B}[/latex]. Ans: 0o
[latex]\vec{A}[/latex] and [latex]\vec{B}[/latex] are two non – zero vectors. If |[latex]\vec{A}\times \vec{B}[/latex]| = [latex]\vec{A}.\vec{B}[/latex], what it the angle between [latex]\vec{A}[/latex] and [latex]\vec{B}[/latex]? Ans: 45o
If [latex]\vec{A}.\vec{B}=0[/latex], what is angle between [latex]\vec{A}[/latex] and [latex]\vec{B}[/latex]? Ans: 90o
Resultant of two equal forces may have the magnitude equal to one of the forces. At what angle between the two equal forces this is possible? Justify your answer. Ans: 120o OR Two vectors have equal magnitudes and their resultant also has the same magnitude. What is the angle between the two vectors?
Two vectors are given as [latex]\vec{V_1} = 2\hat{i}+3\hat{j}+4\hat{k}[/latex] and [latex]\vec{V_2} = 3\hat{i}+2\hat{j}-4\hat{k}[/latex]. Which one of the two is larger in magnitude? Justify your answer. Ans: Equal
A vector is defined as [latex]\vec{E} = 2\hat{i}+3\hat{j}-4\hat{k}[/latex]. What is the magnitude of Y-component of [latex]\vec{E}[/latex]? Ans: 3
Can the sum of two equal vectors be equal to either vector? Ans: Yes, [latex]\theta = 120^o[/latex]
Given two vectors [latex]\vec{A} = 4.00\hat{i}+3.00\hat{j}[/latex] and [latex]\vec{B}=5.00\hat{i}-2.00\hat{j}[/latex]. Find the magnitude of each vector. Ans: A = 5.00, [latex]B=\sqrt{29}[/latex]
A vector [latex]\vec{F}=\hat{i}+2\hat{j}-3\hat{k}[/latex] is given. What is the magnitude of the y-component of the vector? Ans: 2
Is a physical quantity having magnitude and direction necessarily a vector quantity? Explain.
A force (in Newton) expressed in vector notation as [latex]\vec{F}=2\hat{i}+\hat{j}-3\hat{k}[/latex] is applied on a body so that the displacement produced in meter is given by [latex]\vec{D}=\hat{i}-2\hat{j}-3\hat{k}[/latex]. Express the result and nature of the work done. Ans: 9 J
A force (in Newton) expressed in vector notation as [latex]\vec{F}=4\hat{i}+7\hat{j}-3\hat{k}[/latex] is applied on a body and produces a displacement (in meter), [latex]\vec{D}=3\hat{i}-2\hat{j}-5\hat{k}[/latex] in 4 seconds. Estimate the power. Ans: 3.25 watt
What does [latex]\vec{A}.\vec{A}[/latex], the scalar product of a vector with itself give? What about [latex]\vec{A}\times \vec{A}[/latex] the vector product of a vector with itself?
Can you find a vector quantity that has a magnitude of zero but components that are different from zero? Explain.
If [latex]\hat{i}, \hat{j}[/latex] and [latex]\vec{k}[/latex] are unit vectors along x, y and z – axis respectively, find [latex]\hat{i}.(\hat{k}\times \hat{j})[/latex]. Ans: – 1
The angle between two vectors [latex]\vec{A}[/latex] and [latex]\vec{B}[/latex] is θ. Find the magnitude and direction of [latex]\vec{A}\times \vec{B}[/latex] and [latex]\vec{A}.\vec{B}.[/latex]
[latex]\hat{i},\hat{j}[/latex] and [latex]\hat{k}[/latex] are unit vectors of a force along X, Y and Z axes respectively. Find the magnitude and direction of the vector product of two forces [latex]\vec{F_1}[/latex] and [latex]\vec{F_2}[/latex] if [latex]\vec{F_1}=3\hat{i}[/latex] and [latex]\vec{F_2} = -2\hat{k}.[/latex] Ans: [latex]6\hat{j}[/latex]
If [latex]\vec{A}[/latex] and [latex]\vec{B}[/latex] are non zero vectors, is it possible for [latex]\vec{A}\times \vec{B}[/latex] and [latex]\vec{A}.\vec{B}[/latex] both to be zero? Explain. Ans: Impossible
If a vector has zero magnitude, is it meaningful to call it vector? Ans: Yes
If the scalar product of two vectors is equal to the magnitude of their vector product, find the angle between them. Ans: [latex]45^o[/latex]
[latex]\vec{C}[/latex] is the vector sum of [latex]\vec{A}[/latex] and [latex]\vec{B}[/latex] i.e. [latex]\vec{C} = \vec{A} + \vec{B}[/latex] for C = A + B to be true, what is the angle between [latex]\vec{A}[/latex] and [latex]\vec{B}[/latex]? Ans: [latex]0^o[/latex]
The magnitude of two vectors are 3 and 4, and their product is 6. What is the angle between them? Ans: [latex]60^o[/latex]
Two vectors [latex]\vec{A}[/latex] and [latex]\vec{B}[/latex] are such that [latex]\vec{A}-\vec{B} = \vec{C}[/latex] and A – B = C. Find the angle between them. Ans: [latex]0^o[/latex]
What is the difference between scalar and vector products of two vectors? Explain.
If the scalar product of two vectors is equal to the magnitude of their vector product, find the angle between them. Ans: [latex]45^o[/latex]
If [latex]\vec{B}[/latex] is added [latex]\vec{A}[/latex], under what condition does the vector have a magnitude equal to A + B? Under what conditions is the resultant vector equal to zero?
Is momentum of a moving body a vector or a scalar?
Long Questions
State and explain the parallelogram law of vector addition, derive the magnitude and direction of the resultant vector.
State triangle law of vector addition. Obtain an expression for the resultant of two vectors P and Q inclined at angle [latex]\theta[/latex].
Numerical Problems
A disoriented physics professor drives 3.25 km north, then 4.75 km west and then 1.50 km south. Find the magnitude and direction of the resultant displacement. (2076 ‘B’) Ans: 5.06 Km, 159.8o
A rocket fires two engines simultaneously. One produces a thrust of 725 N directly forward, while the other gives a 513 N thrust at 32.4o above the forward direction. Find the magnitude and direction of the resultant force that these engines exert on the rocket. (2075 ‘A’) Ans: 1190 N, 13.4o with[latex]\vec{F_1}[/latex]
