Question

If vector P = ai + aj + 3k and Q = ai - 2j - k are perpendicular to each other, then the positive value of a is

Answer 1

Given,

Two perpendicular vectors P = ai + aj + 3k and Q = ai - 2j - k

To Find,

The positive value of a.

Solution,

Since we know that the dot product of two perpendicular vectors is always 0.

So, the dot product of these two vectors will be

a.a+a(-2)-3 = 0

a²-2a-3 = 0

a²-3a+a-3 = 0

a(a-3)+1(a-3) = 0

(a-3)(a+1) = 0

So,the values of a is 3  and -1.

Hence, the positive value of a is 3.

Answer 2

Answer:

The positive value of $a$ is $3$.

Explanation:

• The dot product is zero for the perpendicular vectors.
• The individual components are multiplied and then added when we calculate the dot product of two vectors.
• The cross product gives a vector term.

We know that the two vectors, $P$ and $Q$ are perpendicular to each other.

So, $P\cdot Q=0$

$(ai + aj + 3k)\cdot (ai - 2j - k)=0$

$a^{2} -2a-3=0$

Solving the above quadratic equation to get the value of $a$:

$a^{2} -3a + a-3=0$

$a(a-3)+1(a-3)=0$

$(a+1)(a-3)=0$

$a=3$ or $a=-1$

Hence, the positive value of $a$ is $3$.