Vector Algebra

Ex 10.1

Question 1:

Represent graphically a displacement of 40 km, 30° east of north.

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Answer:

Here, vector represents the displacement of 40 km, 30° East of North.

 Question 2:

Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 metres north-west (iii) 40°

(iv) 40 watt (v) 10^–19 coulomb (vi) 20 m/s²

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Answer:

(i) 10 kg is a scalar quantity because it involves only magnitude.

(ii) 2 meters north-west is a vector quantity as it involves both magnitude and direction.

(iii) 40° is a scalar quantity as it involves only magnitude.

(iv) 40 watts is a scalar quantity as it involves only magnitude.

(v) 10^–19 coulomb is a scalar quantity as it involves only magnitude.

(vi) 20 m/s² is a vector quantity as it involves magnitude as well as direction.

 

Question 3:

Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

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Answer:

(i) Time period is a scalar quantity as it involves only magnitude.

(ii) Distance is a scalar quantity as it involves only magnitude.

(iii) Force is a vector quantity as it involves both magnitude and direction.

(iv) Velocity is a vector quantity as it involves both magnitude as well as direction.

(v) Work done is a scalar quantity as it involves only magnitude.

Question 4:

In Figure, identify the following vectors.

(i) Coinitial (ii) Equal (iii) Collinear but not equal

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Answer:

(i) Vectors and are coinitial because they have the same initial point.

(ii) Vectorsandare equal because they have the same magnitude and direction.

(iii) Vectorsand are collinear but not equal. This is because although they are parallel, their directions are not the same.

Question 5:

Answer the following as true or false.

(i)  andare collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

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Answer:

(i) True.

Vectors  andare parallel to the same line.

(ii) False.

Collinear vectors are those vectors that are parallel to the same line.

(iii) False.

It is not necessary for two vectors having the same magnitude to be parallel to the same line.

(iv) False.

Two vectors are said to be equal if they have the same magnitude and direction, regardless of the positions of their initial points.

Ex10.2

Question 1:

Compute the magnitude of the following vectors:

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Answer:

The given vectors are:

Question 2:

Write two different vectors having same magnitude.

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Answer:

Hence, are two different vectors having the same magnitude. The vectors are different because they have different directions.

Question 3:

Write two different vectors having same direction.

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Answer:

The direction cosines of are the same. Hence, the two vectors have the same direction.

Question 4:

Find the values of x and y so that the vectors are equal

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Answer:

The two vectors will be equal if their corresponding components are equal.

Hence, the required values of x and y are 2 and 3 respectively.

Question 5:

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

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Answer:

The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by,

Hence, the required scalar components are –7 and 6 while the vector components are 

Question 6:

Find the sum of the vectors.

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Answer:

The given vectors are.

Question 7:

Find the unit vector in the direction of the vector.

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Answer:

The unit vector in the direction of vector is given by.