Question

Calculate the values of :
i) j cap . (2i cap - 3 j cap + k cap)
ii) (2 i cap=j cap) . (3 i cap +k cap)
Please fast

Answer 1

1 -:
Jcap .(2icap-3jcap+kcap)
(0icap + jcap + 0kcap )(2icap-3jcap+kcao)
0-3+0 = 3
2-:
(2icap-jcap)(3icap+kcap) =6+0+0 =6

Answer 2

Answer:

(i) j . (2i - 3 j + k) = -3.

(ii) (2 i - j) . (3 i  + k) = 6.  

Explanation:

• A vector represents a physical quantity that has both magnitude and direction such as displacement, velocity etc.

• Unit vectors i, j, k: These are the vectors that have a magnitude of 1 and their directions align with the direction of X, Y, Z axes.
• Product of two vectors is of two types :

1. Dot product - Also called the scalar product as the resultant of the product is a scalar quantity. Dot product of 2 vectors a, b is calculated as a.b = |a||b|cosθ.                                                                 where |a|, |b| are magnitudes of vectors a, b  and                                     θ is the angle between the vectors a,b.                                            
2. Cross product - Also called the vector product as the resultant of the product is a vector quantity. Cross product of 2 vectors a, b is calculated as a.b = |a||b|sinθ.                                                                where |a|, |b| are magnitudes of vectors a, b  and                                     θ is the angle between the vectors a,b.  

Calculation:

The magnitudes of i, j, k are 1 and the angle between any two of the three vectors is 90°.

i.i = 1*1*cos 0° = 1

i.j = j.i = j.k = k.j = i.k = k.i = 1*1*cos 90° = 0

(i) j  . (2i - 3 j + k)

= 2*j.i - 3*j.j + k.j

= 2*0 - 3*1 + 0

= -3

Therefore,  j . (2i - 3 j + k) = -3

(ii) (2 i - j) . (3 i  + k)

= 2i.3i - j.3i + 2i.k - j.k

= 6 - 0 + 0 - 0

= 6

Therefore, (2 i - j) . (3 i  + k) = 6.

Find more about product of vectors:

Scalar and Vector Product of two vectors

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Properties of vector product

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