If a ^ – b ^ = 2 then calculate the value of a ^ + 3 b ^ .

Given, a ^ – b ^ = 2 , where a ^ and b ^ are unit vectors. Taking squares both sides, a ^ – b ^ 2 = 2 2 a ^ 2 + b ^ 2 - 2 a ^ b ^ cos θ = 2 ( θ is the angle between the two unit vectors) 1 2 + 1 2 - 2 × 1 × 1 × cos θ = 2 cos θ = 0 or θ = 90 ° Now, let's calculate a ^ + 3 b ^ Solve the expression by taking square and under root at the same time, a ^ + 3 b ^ 2 = a ^ 2 + 3 b ^ 2 + 2 a ^ 3 b ^ cos 90 ° = 1 2 + 3 2 + 2 × 3 × cos 90 ° = 2

If A → = 3 i ^ + 2 j ^ and B → = 2 i ^ + 3 j ^ – k ^ , then find a unit vector along A → – B → .

Given, two vectors as A → = 3 i ^ + 2 j ^ and B → = 2 i ^ + 3 j ^ – k ^ . Now, A → – B → = 3 i ^ + 2 j ^ - 2 i ^ + 3 j ^ – k ^ A → – B → = i ^ – j ^ + k ^ Unit vector of any vector P → can be find out as, P ^ = P → P → We know that the magnitude of a vector P → = x i ^ + y j ^ + z k ^ is given by, P → = x 2 + y 2 + z 2 . So, unit vector of A → – B → is i ^ – j ^ + k ^ 1 2 + 1 2 + 1 2 = i ^ – j ^ + k ^ 3

If magnitude of sum of two unit vectors is 2 then find the magnitude of subtraction of these unit vectors.

Suppose a ^ and b ^ are the unit vectors given. It is given that a ^ + b ^ = 2 Taking squares both sides, a ^ + b ^ 2 = 2 2 a ^ 2 + b ^ 2 + 2 a ^ b ^ cos θ = 2 ( θ is the angle between the two unit vectors) 1 2 + 1 2 + 2 × 1 × 1 × cos θ = 2 cos θ = 0 or θ = 90 ° Now, let's calculate a ^ - b ^ Solve the expression by taking square and under root at the same time, a ^ - b ^ 2 = a ^ 2 + b ^ 2 + 2 a ^ b ^ cos 90 ° = 1 2 + 1 2 + 2 × 3 × cos 90 ° = 2

The x and y components of vector A → are 4 m and 6 m respectively. The x and y components of vector A → + B → are 10 m and 9 m respectively. For the vector B → calculate the following x and y components :

Given x component of A → is 4m and y component is 6m respectively. And, A → + B → has its x component as 10m and y component as 9m respectively. Therefore, the x component of B → is (10-4)m and the y component is (9-6)m respectively. The x and y components are 6m and 3m respectively.

The x and y components of vector A → are 4 m and 6 m respectively. The x and y components of vector A → + B → are 10 m and 9 m respectively. For the vector B → calculate the following length :

Given x component of A → is 4m and y component is 6m respectively. And, A → + B → has its x component as 10m and y component as 9m respectively. Therefore, the x component of B → is (10-4)m and the y component is (9-6)m respectively. The x and y components are 6m and 3m respectively. So the resultant length of vector b = 6 2 + 3 2 = 45

The x and y components of vector A → are 4 m and 6 m respectively. The x and y components of vector A → + B → are 10 m and 9 m respectively. For the vector B → calculate the following the angle it makes with x -axis :

Given x component of A → is 4m and y component is 6m respectively. And, A → + B → has its x component as 10m and y component as 9m respectively. Therefore, the x component of B → is (10-4)m and the y component is (9-6)m respectively. The x and y components are 6m and 3m respectively. If ∅ is the angle that the resultant makes with x-axis and the vector can be written as x i ^ + y j ^ Then, tan ∅ = y x Therefore, here tan ∅ = 3 6 = 1 2

Find the directional cosines of vector 5 i ^ + 2 j ^ + 6 k ^ . Also write the value of sum of squares of directional cosines of this vector.

If a vector P → = x i ^ + y j ^ + z k ^ makes angle with X , Y and Z axis as α , β and γ then direction cosines of the vector are defined as cos α = x x 2 + y 2 + z 2 , cos β = y x 2 + y 2 + z 2 and cos γ = z x 2 + y 2 + z 2 If the vector is given as 5 i ^ + 2 j ^ + 6 k ^ then put the values of the components in the above equations, cos α = 5 5 2 + 2 2 + 6 2 = 5 65 cos β = 2 5 2 + 2 2 + 6 2 = 2 65 cos γ = 6 5 2 + 2 2 + 6 2 = 6 65 Now, the value of sum of squares of direction cosines are cos 2 α + cos 2 β + cos 2 γ 5 65 2 + 2 65 2 + 6 65 2 = 1

If A → = 6 i ^ – 6 j ^ + 5 k ^ and B → = i ^ + 2 j ^ – k ^ , then find a unit vector parallel to the resultant of A → & B → .

Given, two vectors as A → = 6 i ^ – 6 j ^ + 5 k ^ and B → = i ^ + 2 j ^ – k ^ . Now, resultant of both vectors, i.e., A → + B → = 6 i ^ – 6 j ^ + 5 k ^ + i ^ + 2 j ^ – k ^ A → + B → = 7 i ^ – 4 j ^ + 4 k ^ Unit vector of any vector P → can be find out as, P ^ = P → P → We know that the magnitude of a vector P → = x i ^ + y j ^ + z k ^ is given by, P → = x 2 + y 2 + z 2 . So, unit vector of A → + B → is 7 i ^ – 4 j ^ + 4 k ^ 7 2 + 4 2 + 4 2 = i ^ – j ^ + k ^ 9

E M B I B E

Beta Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise (Conceptual Questions)>Q 10

EASY

NEET

IMPORTANT

Earn 100

If two forces act in opposite direction then their resultant is $10 N$ and if they act mutually perpendicular then their resultant is $50 N$ . Find the magnitudes of both the forces.

Important Questions on Mathematical Tools in Physics

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IMPORTANT

Beta Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise (Conceptual Questions)>Q 11

If

|\hat{a} - \hat{b}| = \sqrt{2}

then calculate the value of

|\hat{a} + \sqrt{3} \hat{b}|

.

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IMPORTANT

Beta Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise (Conceptual Questions)>Q 12

If

\vec{A} = 3 \hat{i} + 2 \hat{j}

and

\vec{B} = 2 \hat{i} + 3 \hat{j} - \hat{k}

, then find a unit vector along

(\vec{A} - \vec{B})

.

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IMPORTANT

Beta Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise (Conceptual Questions)>Q 13

If magnitude of sum of two unit vectors is

\sqrt{2}

then find the magnitude of subtraction of these unit vectors.

EASY

NEET

IMPORTANT

Beta Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise (Conceptual Questions)>Q 14

The $x$ and $y$ components of vector $\vec{A}$ are $4 m$ and $6 m$ respectively. The $x$ and $y$ components of vector $B = 30 N$ are $10 m$ and $9 m$ respectively. For the vector $\vec{B}$ calculate the following