How many of the following can be a vector perpendicular to both the vectors \(2\hat{i} - \hat{j} + \hat{k}\) and \(\hat{i} + \hat{j} + 3\hat{k}\) ? I. \(4\hat{i} + 5\hat{j} - 3\hat{k}\) II. \(-8\hat{i} - 10\hat{j} + 6\hat{k}\) III. \(\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})\) Select the correct answer.
All three
To determine how many of the given vectors can be perpendicular to both vectors \(2\hat{i} - \hat{j} + \hat{k}\) and \(\hat{i} + \hat{j} + 3\hat{k}\), we need to check the orthogonality of each option with both these vectors.
A vector \(\mathbf{v}\) is perpendicular to vectors \(\mathbf{a}\) and \(\mathbf{b}\) if the dot product \(\mathbf{v} \cdot \mathbf{a} = 0\) and \(\mathbf{v} \cdot \mathbf{b} = 0\).
- Vector I: \(4\hat{i} + 5\hat{j} - 3\hat{k}\)
- Calculate the dot product with \(2\hat{i} - \hat{j} + \hat{k}\): \((\mathbf{v}_1 \cdot \mathbf{a}) = 4 \times 2 + 5 \times (-1) + (-3) \times 1 = 8 - 5 - 3 = 0\)
- Calculate the dot product with \(\hat{i} + \hat{j} + 3\hat{k}\): \((\mathbf{v}_1 \cdot \mathbf{b}) = 4 \times 1 + 5 \times 1 + (-3) \times 3 = 4 + 5 - 9 = 0\)
- The vector is perpendicular to both vectors.
- Vector II: \(-8\hat{i} - 10\hat{j} + 6\hat{k}\)
- Calculate the dot product with \(2\hat{i} - \hat{j} + \hat{k}\): \((\mathbf{v}_2 \cdot \mathbf{a}) = (-8) \times 2 + (-10) \times (-1) + 6 \times 1 = -16 + 10 + 6 = 0\)
- Calculate the dot product with \(\hat{i} + \hat{j} + 3\hat{k}\): \((\mathbf{v}_2 \cdot \mathbf{b}) = (-8) \times 1 + (-10) \times 1 + 6 \times 3 = -8 - 10 + 18 = 0\)
- The vector is perpendicular to both vectors.
- Vector III: \(\frac{1}{50}(-4\hat{i} - 5\hat{j} + 3\hat{k})\)
- Basically considering the vector without scale as \((-4\hat{i} - 5\hat{j} + 3\hat{k})\)
- Calculate the dot product with \(2\hat{i} - \hat{j} + \hat{k}\): \((\mathbf{v}_3 \cdot \mathbf{a}) = (-4) \times 2 + (-5) \times (-1) + 3 \times 1 = -8 + 5 + 3 = 0\)
- Calculate the dot product with \(\hat{i} + \hat{j} + 3\hat{k}\): \((\mathbf{v}_3 \cdot \mathbf{b}) = (-4) \times 1 + (-5) \times 1 + 3 \times 3 = -4 - 5 + 9 = 0\)
- The vector is perpendicular to both vectors.
Since all three vectors satisfy the conditions of being perpendicular to both vectors \(2\hat{i} - \hat{j} + \hat{k}\) and \(\hat{i} + \hat{j} + 3\hat{k}\), the correct answer is All three.
Similar Questions
Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-zero vectors such that \(\vec{a}\times \vec{b} = \vec{c} \) . Consider the following statements:
1. \(\vec a\) is unique if \(\vec b\) and \(\vec c\) are given
2. \(\vec c\) is unique if \(\vec a\) and \(\vec b\) are given
Which of the above statements is/are correct?
In a right angled triangle ABC, if the hypotenuse AC = p, then what is \(\overrightarrow {{\rm{AB}}} \cdot \overrightarrow {{\rm{AC}}} + \overrightarrow {{\rm{BC}}} \cdot \overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CA}}} \cdot \overrightarrow {{\rm{CB}}} \) equal to?
What is \({\rm{\vec c}}\) equal to?
If \({\rm{\vec d}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\) , then which of the following equations is/are correct?
1. y – x = 4
2. 2z – 3 = 0
Select the correct answer using the code given below:
What is \({\rm{\vec a}} \cdot {\rm{\vec b}} + {\rm{\vec b}} \cdot {\rm{\vec c}} + {\rm{\vec c}} \cdot {\rm{\vec a}}\) equal to?
What is the angle between \({\rm{\vec a}}\) and \({\rm{\vec b}}\) ?
- What is the fourth vertex \(D\)?
- Let \(\vec{a} = \hat{i} - \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\). If \(\vec{a} \times (\vec{b} \times \vec{a}) = \alpha\hat{i} - \beta\hat{j} + \gamma\hat{k}\), then what is the value of \(\alpha + \beta + \gamma\)?
- Let \(\vec{p}=\vec{a}-\vec{b}\), \(\vec{q}=\vec{a}+\vec{b}\). If \(|\vec{a}|=|\vec{b}|= 2\) and \(\vec{a}\cdot\vec{b} = 2\), then what is the value of \(|\vec{p}\times\vec{q}|\)?
- What is the area of the parallelogram whose sides are represented by the vectors \(\hat{i} + 2\hat{j} + 3\hat{k}\) and \(2\hat{i} + \hat{j} + 2\hat{k}\)?
Important Questions from Vector Algebra
Vector A̅ = ŷ.3 + ẑ.2 and B̅ = x̂.5 + ŷ.8 extend from the origin. Find A̅.B̅ Choose the correct answer.
The value of the cross product \(\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)\) of two vectors \(\overrightarrow a - \overrightarrow b\) and \(\overrightarrow a + \overrightarrow b \) is:
If non - zero a, b, c are such that a + b + c = 0, then the value of \(\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab}\) is
Vector a = 3i + 2j – 6k, vector b = 4i – 3j + k, angle between above vectors is
Two forces F1 and F2 are used to pull a car, which met an accident. The angle between the two force is θ. Find the value of θ for the resultant force is equal to \(\sqrt{(F_1^2 + F_2^2)}\)
